101098rspa20010888

Huygensrsquos clocks

By Matthew Bennett1 Michael F Schatz1Heidi Rockwood2 a nd Kurt Wiesenfeld1

1Center for Nonlinear Science and School of Physics and2School of Modern Languages Georgia Institute of Technology

Atlanta GA 30332-0430 USA

Received 3 May 2001 revised 29 June 2001 accepted 13 July 2001published online 25 January 2002

The 336-year-old synchronization observations of Christiaan Huygens are re-exam-ined in modern experiments A simple model of synchronization is proposed

Keywords synchronization nonlinear dynamics coupled oscillators

1 Introduction

Shortly after The Royal Societyrsquos founding in 1660 Christiaan Huygens in partner-ship with the Society set out to solve the outstanding technological challenge of theday the longitude problem ie shy nding a robust accurate method of determining lon-gitude for maritime navigation (Yoder 1990) Huygens had invented the pendulumclock in 1657 (Burke 1978) and subsequently had demonstrated mathematicallythat a pendulum would follow an isochronous path independent of amplitude ifcycloidal-shaped plates were used to conshy ne the pendulum suspension (Yoder 1990)Huygens believed that cycloidal pendulum clocks suitably modishy ed to withstandthe rigours of sea travel could provide timing of sumacr cient accuracy to determinelongitude reliably Maritime pendulum clocks were constructed by Huygens in col-laboration with one of the original fellows of The Royal Society Alexander Bruce2nd Earl of Kincardine Over the course of three years (16621665) Bruce and theSociety supervised sea trials of the clocks Meanwhile Huygens remaining in TheHague continually corresponded with the Society through Sir Robert Moray bothto inquire about the outcome of the sea trials and to describe the ongoing enotortsHuygens was making to perfect the design of maritime clocks On 1 March 1665Moray read to the Society a letter from Huygens dated 27 February 1665 reportingof (Birch 1756)

an odd kind of sympathy perceived by him in these watches [two maritimeclocks] suspended by the side of each other

Huygensrsquos study of two clocks operating simultaneously arose from the practicalrequirement of redundancy for maritime clocks if one clock stopped (or had to becleaned) then the other could be used to provide timekeeping (Huygens 1669) Ina contemporaneous letter to his father Huygens further described his observationsmade while conshy ned to his rooms by a brief illness Huygens found that the pendulumclocks swung in exactly the same frequency and 180macr out of phase (Huygens 1950a b)

Proc R Soc Lond A (2002) 458 563579

563

creg 2002 The Royal Society

564 M Bennett and others

When he disturbed one pendulum the anti-phase state was restored within half anhour and remained indeshy nitely

Motivated by the belief that synchronization could be used to keep sea clocks inprecise agreement (Yoder 1990) Huygens carried out a series of experiments in anenotort to understand the phenomenon He found that synchronization did not occurwhen the clocks were removed at a distance or oscillated in mutually perpendicularplanes Huygens deduced that the crucial interaction came from very small move-ments of the common frame supporting the two clocks He also provided a physicalexplanation for how the frame motion set up the anti-phase motion but though hisprowess was great his tools were limited his discovery of synchronization occurredin the same year when young Isaac Newton removed to his country home to escapethe Black Plague and begin the work that eventually led to his Principia publishedsome 20 years later

The Royal Society viewed Huygensrsquos explanation of synchronization as a setbackfor using pendulum clocks to determine longitude at sea (Birch 1756)

Occasion was taken here by some of the members to doubt the exactnessof the motion of these watches at sea since so slight and almost insensiblemotion was able to cause an alteration in their going

Ultimately the innovation of the pendulum clock did not solve the longitude prob-lem (Britten 1973) However Huygensrsquos synchronization observations have servedto inspire study of sympathetic rhythms of interacting nonlinear oscillators in manyareas of science The onset of synchronization and the selection of particular phaserelations is a fundamental problem of nonlinear dynamics and one which has beenavidly pursued in recent years in problems ranging from neurobiology and brain func-tion (Rodriguez et al 1999) to animal locomotion (Strogatz amp Stewart 1993 Gol-ubitsky et al 1999) superconducting electronics laser physics and smart antennaarrays (Liao amp York 1993)

In this paper we reconsider Huygensrsquos observations To our knowledge previousattempts to understand Huygensrsquos observations are few and ultimately unsatisfac-tory We have built an updated version of the two-clock system with pendulumsattached to a common frame free to move in one dimension In our experiments wevary the coupling strength by changing the ratio of pendulum mass to system mass middot and thereby explore in greater depth the situation facing Huygens At small middot (weakcoupling) corresponding to Huygensrsquos situation we shy nd that whenever the pendu-lums frequency lock they fall into anti-phase oscillations However as the coupling isincreased by increasing middot we observe another state in which one or both clocks ceaseto run a state we call `beating deathrsquo This behaviour is increasingly dominant as middotbecomes large Thus our results suggest Huygensrsquos observations depended somewhatserendipitously on the extra heavy weighting of his clocks intended to make themmore stable at sea

We study the problem theoretically by deriving a Poincare map for the nonlineardynamics Our map is in agreement with the experimental observations moreover inone useful limit the map reduces to a single degree of freedom and captures manyessential results of our experiments We are also able to explain the behaviour invery direct physical terms based on a normal mode description a picture originallyput forward by Korteweg (1906)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 565

2 Background

(a) Details of Huygensrsquos clock experiments

The operation of a pendulum clock is described in great detail by Huygens (Huygens1986) In brief the pendulum is attached to an escapement which alternately blocksand releases a scape wheel as the pendulum oscillates This action provides the timingthat is transmitted from the scape wheel to the clocksrsquo hands via a gearing system(the motion work ) The scape wheel is also connected through a separate gearingsystem (the going train) to an energy source typically a wound spring or elevatedweights The unwinding spring or falling weights drive the motion of the scape wheelwhich in turn provides small `kicksrsquo to the pendulum via the escapement Thistransmission of energy to the pendulum compensates for losses due to friction thusthe pendulum continues to oscillate indeshy nitely as long as the spring is periodicallyrewound or the fallen weights are periodically raised back up

Many experimental details of Huygensrsquos observations are recorded in his writings(Huygens 1950a b Huygens 1986) The pendulum in each clock measured ca 9 iny inlength corresponding to an oscillation period of ca 1 s Each pendulum weighed 1

2lbz

and regulated the clock through a verge escapement which required each pendulumto execute large angular displacement amplitudes of ca 20macr or more from verticalfor the clock to function (see Rawlings (1944) and also Landes (1983 appendix A)for a detailed description of the verge escapement) The amplitude dependence ofthe period in these clocks was typically corrected by use of cycloidal-shaped bound-aries to conshy ne the suspension (Huygens 1986) Each pendulum clock was enclosedin a 4 ft long case a weight of ca 100 lb was placed at the bottom of each case(to keep the clock oriented aboard a ship) From this information we estimate thatthe important experimental parameter middot the ratio of the single pendulum mass tothe total system mass is ca 0005 Though the two clocks dinotered in certain phys-ical aspects (for example Huygens notes that the size of the clocks was somewhatdinoterent (Huygens 1950a b)) the clocks were closely matched in those characteris-tics we expect are essential for the dynamics In particular well-adjusted pendulumclocks of the 1660s would typically run at rates which dinotered by only 15 s per day(so their natural frequencies dinotered by approximately two parts in 10 000) (Landes1983) Pendulum clocks represented a tremendous advance in horology the science oftimekeeping Before Huygensrsquos invention of the pendulum clock typical clocks (egverge escapement with balance wheel regulator) varied by ca 15 min (1) per day(Landes 1983)

Huygensrsquos laboratory notebook contains a detailed description of tests and observa-tions on synchronization (Huygens 1950a b) In some experiments Huygens studiedtwo clocks that were suspended side by side each hanging from a hook embeddedin the same wooden beam In other experiments Huygens studied a conshy gurationwith each clock hanging from its own wooden beam and the two beams lying ontop of back to back chairs At shy rst Huygens suspected the `sympathyrsquo was dueto induced air currents but eventually concluded the cause was the `imperceptiblemovementsrsquo of the common supporting structure The associated coupling was weakwhen Huygens disturbed the pendulums he found the clocks required ca 30 min

y 1 in equals 254 cmz 1 lb equals 04536 kg 1 ft equals 12 in equals 305 cm

Proc R Soc Lond A (2002)

566 M Bennett and others

(a)

(b)

(1) (2)

Figure 1 Multiple-exposure images illustrate the anti-phase attractor of a large mass ratio( middot = 00063) system of two pendulums mounted on a common translating beam (a) The systemis started with equal amplitude in-phase oscillation of the left (1) and the right (2) pendulumsIn the presence of weak coupling through lateral motion of the mounting cart the in-phase stateis unstable and after a suplusmn cient time the system oscillates stably in anti-phase macrnal state (b)

before synchronization was restored At one second per cycle a transient time of30 min amounts to 1800 cycles (Interestingly Huygens considers this restorationtime as fast perhaps a reregection of the pace of life then and now) The dissipationwas also weak as can be seen from the following estimate the total energy avail-able to run each clock is ca 100 J obtained from weights of ca 10 kg mass fallinga distance of 1 m Assuming this energy is sumacr cient to keep a clock running forca 250 000 oscillations (ca 3 days) the energy input per oscillation is ca 4 pound 10iexcl4 JThe total energy in a 9 in long 1

2lb pendulum swinging through a semiarc of 25macr

is ca 005 J Thus energy loss per oscillation relative to the pendulum energy isca 08

(b) Recapping Korteweg and Blekhman studies

We know of two studies which were directly motivated by Huygensrsquos observationsThe shy rst was Kortewegrsquos (1906) paper in which he analysed a three-degree-of-freedommodel consisting of two plane pendulums connected to a rigid frame free to oscillatein one dimension (Korteweg was also strongly motivated by the 18th century obser-vations of Ellicott regarding the slow beating between weakly coupled pendulums)Korteweg made a linear normal mode analysis for small oscillations in the absenceof damping and driving enotects To explain why only certain of these modes might beobserved and others not Korteweg introduced the idea that friction is responsiblefor certain motions being unsustainable If a mode involved large-amplitude motionof the supporting frame he argued the internal clock mechanisms would be unableto provide enough energy to sustain this motion Conversely for a mode in whichthe frame moved only a little the energy input could overcome the enotects of friction

Proc R Soc Lond A (2002)

Huygensrsquos clocks 567

0 1000089

094

04

07

10

0

p 2

p

t

A

A1

A2

a(r

ad)

0 1000t

(a) (b)

(c) (d)

Figure 2 (a) (b) Time-series from pendulum experiments and (c) (d) simulations illustratingthe evolution from unstable in-phase to stable anti-phase oscillations Damped beating is foundin both the pendulumsrsquo (a) (c) phase direg erence not and (b) (d) amplitudes A1 A2 Time isscaled by the pendulum period and amplitudes are scaled by the initial amplitude at t = 0 Theexperimental conditions correspond to those listed in macrgure 1

Though these aspects of energy damping and energy input were not included in hisquantitative analysis Korteweg concluded that Huygensrsquos observations were entirelycaptured by the three-degree-of-freedom model and that the anti-phase mode if notthe only sustainable motion enjoyed a distinct advantage over in-phase motion Thisadvantage was cleverly used in a dinoterent context for precise measurements of theacceleration of gravity by Vening Meinesz and others who used pairs of pendulumsset into free (no driving) anti-phase oscillation (Heiskanen amp Vening Meinesz 1958)

Blekhman (1988) also discusses Huygensrsquos observations in his book and he re-counts the results of a laboratory reproduction of the coupled clocks as well as pre-senting a theoretical analysis of oscillators coupled through a common supportingframe His model is similar to the one studied by Korteweg except that Blekhmanuses van der Pol oscillators rather than pendulums so that his quantitative analysisincludes both (weak) driving and damping enotects Blekhman was primarily moti-vated by the general aspects of synchronization phenomena found in a wide vari-ety of physical systems and it is probably for this reason that he chose to usevan der Pol oscillators He predicted that both in-phase and anti-phase motionsare stable under the same circumstances (that is the two are coexisting attrac-tors) Somewhat puzzling is that he reports observing both states in the exper-imental reproduction while this agrees with his predictions for the van der Polsystem Huygens (so far as we know) never mentioned stable in-phase synchroniza-tion

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

564 M Bennett and others

When he disturbed one pendulum the anti-phase state was restored within half anhour and remained indeshy nitely

Motivated by the belief that synchronization could be used to keep sea clocks inprecise agreement (Yoder 1990) Huygens carried out a series of experiments in anenotort to understand the phenomenon He found that synchronization did not occurwhen the clocks were removed at a distance or oscillated in mutually perpendicularplanes Huygens deduced that the crucial interaction came from very small move-ments of the common frame supporting the two clocks He also provided a physicalexplanation for how the frame motion set up the anti-phase motion but though hisprowess was great his tools were limited his discovery of synchronization occurredin the same year when young Isaac Newton removed to his country home to escapethe Black Plague and begin the work that eventually led to his Principia publishedsome 20 years later

The Royal Society viewed Huygensrsquos explanation of synchronization as a setbackfor using pendulum clocks to determine longitude at sea (Birch 1756)

Occasion was taken here by some of the members to doubt the exactnessof the motion of these watches at sea since so slight and almost insensiblemotion was able to cause an alteration in their going

Ultimately the innovation of the pendulum clock did not solve the longitude prob-lem (Britten 1973) However Huygensrsquos synchronization observations have servedto inspire study of sympathetic rhythms of interacting nonlinear oscillators in manyareas of science The onset of synchronization and the selection of particular phaserelations is a fundamental problem of nonlinear dynamics and one which has beenavidly pursued in recent years in problems ranging from neurobiology and brain func-tion (Rodriguez et al 1999) to animal locomotion (Strogatz amp Stewart 1993 Gol-ubitsky et al 1999) superconducting electronics laser physics and smart antennaarrays (Liao amp York 1993)

In this paper we reconsider Huygensrsquos observations To our knowledge previousattempts to understand Huygensrsquos observations are few and ultimately unsatisfac-tory We have built an updated version of the two-clock system with pendulumsattached to a common frame free to move in one dimension In our experiments wevary the coupling strength by changing the ratio of pendulum mass to system mass middot and thereby explore in greater depth the situation facing Huygens At small middot (weakcoupling) corresponding to Huygensrsquos situation we shy nd that whenever the pendu-lums frequency lock they fall into anti-phase oscillations However as the coupling isincreased by increasing middot we observe another state in which one or both clocks ceaseto run a state we call `beating deathrsquo This behaviour is increasingly dominant as middotbecomes large Thus our results suggest Huygensrsquos observations depended somewhatserendipitously on the extra heavy weighting of his clocks intended to make themmore stable at sea

We study the problem theoretically by deriving a Poincare map for the nonlineardynamics Our map is in agreement with the experimental observations moreover inone useful limit the map reduces to a single degree of freedom and captures manyessential results of our experiments We are also able to explain the behaviour invery direct physical terms based on a normal mode description a picture originallyput forward by Korteweg (1906)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 565

2 Background

(a) Details of Huygensrsquos clock experiments

The operation of a pendulum clock is described in great detail by Huygens (Huygens1986) In brief the pendulum is attached to an escapement which alternately blocksand releases a scape wheel as the pendulum oscillates This action provides the timingthat is transmitted from the scape wheel to the clocksrsquo hands via a gearing system(the motion work ) The scape wheel is also connected through a separate gearingsystem (the going train) to an energy source typically a wound spring or elevatedweights The unwinding spring or falling weights drive the motion of the scape wheelwhich in turn provides small `kicksrsquo to the pendulum via the escapement Thistransmission of energy to the pendulum compensates for losses due to friction thusthe pendulum continues to oscillate indeshy nitely as long as the spring is periodicallyrewound or the fallen weights are periodically raised back up

Many experimental details of Huygensrsquos observations are recorded in his writings(Huygens 1950a b Huygens 1986) The pendulum in each clock measured ca 9 iny inlength corresponding to an oscillation period of ca 1 s Each pendulum weighed 1

2lbz

and regulated the clock through a verge escapement which required each pendulumto execute large angular displacement amplitudes of ca 20macr or more from verticalfor the clock to function (see Rawlings (1944) and also Landes (1983 appendix A)for a detailed description of the verge escapement) The amplitude dependence ofthe period in these clocks was typically corrected by use of cycloidal-shaped bound-aries to conshy ne the suspension (Huygens 1986) Each pendulum clock was enclosedin a 4 ft long case a weight of ca 100 lb was placed at the bottom of each case(to keep the clock oriented aboard a ship) From this information we estimate thatthe important experimental parameter middot the ratio of the single pendulum mass tothe total system mass is ca 0005 Though the two clocks dinotered in certain phys-ical aspects (for example Huygens notes that the size of the clocks was somewhatdinoterent (Huygens 1950a b)) the clocks were closely matched in those characteris-tics we expect are essential for the dynamics In particular well-adjusted pendulumclocks of the 1660s would typically run at rates which dinotered by only 15 s per day(so their natural frequencies dinotered by approximately two parts in 10 000) (Landes1983) Pendulum clocks represented a tremendous advance in horology the science oftimekeeping Before Huygensrsquos invention of the pendulum clock typical clocks (egverge escapement with balance wheel regulator) varied by ca 15 min (1) per day(Landes 1983)

Huygensrsquos laboratory notebook contains a detailed description of tests and observa-tions on synchronization (Huygens 1950a b) In some experiments Huygens studiedtwo clocks that were suspended side by side each hanging from a hook embeddedin the same wooden beam In other experiments Huygens studied a conshy gurationwith each clock hanging from its own wooden beam and the two beams lying ontop of back to back chairs At shy rst Huygens suspected the `sympathyrsquo was dueto induced air currents but eventually concluded the cause was the `imperceptiblemovementsrsquo of the common supporting structure The associated coupling was weakwhen Huygens disturbed the pendulums he found the clocks required ca 30 min

y 1 in equals 254 cmz 1 lb equals 04536 kg 1 ft equals 12 in equals 305 cm

Proc R Soc Lond A (2002)

566 M Bennett and others

(a)

(b)

(1) (2)

Figure 1 Multiple-exposure images illustrate the anti-phase attractor of a large mass ratio( middot = 00063) system of two pendulums mounted on a common translating beam (a) The systemis started with equal amplitude in-phase oscillation of the left (1) and the right (2) pendulumsIn the presence of weak coupling through lateral motion of the mounting cart the in-phase stateis unstable and after a suplusmn cient time the system oscillates stably in anti-phase macrnal state (b)

before synchronization was restored At one second per cycle a transient time of30 min amounts to 1800 cycles (Interestingly Huygens considers this restorationtime as fast perhaps a reregection of the pace of life then and now) The dissipationwas also weak as can be seen from the following estimate the total energy avail-able to run each clock is ca 100 J obtained from weights of ca 10 kg mass fallinga distance of 1 m Assuming this energy is sumacr cient to keep a clock running forca 250 000 oscillations (ca 3 days) the energy input per oscillation is ca 4 pound 10iexcl4 JThe total energy in a 9 in long 1

2lb pendulum swinging through a semiarc of 25macr

is ca 005 J Thus energy loss per oscillation relative to the pendulum energy isca 08

(b) Recapping Korteweg and Blekhman studies

We know of two studies which were directly motivated by Huygensrsquos observationsThe shy rst was Kortewegrsquos (1906) paper in which he analysed a three-degree-of-freedommodel consisting of two plane pendulums connected to a rigid frame free to oscillatein one dimension (Korteweg was also strongly motivated by the 18th century obser-vations of Ellicott regarding the slow beating between weakly coupled pendulums)Korteweg made a linear normal mode analysis for small oscillations in the absenceof damping and driving enotects To explain why only certain of these modes might beobserved and others not Korteweg introduced the idea that friction is responsiblefor certain motions being unsustainable If a mode involved large-amplitude motionof the supporting frame he argued the internal clock mechanisms would be unableto provide enough energy to sustain this motion Conversely for a mode in whichthe frame moved only a little the energy input could overcome the enotects of friction

Proc R Soc Lond A (2002)

Huygensrsquos clocks 567

0 1000089

094

04

07

10

0

p 2

p

t

A

A1

A2

a(r

ad)

0 1000t

(a) (b)

(c) (d)

Figure 2 (a) (b) Time-series from pendulum experiments and (c) (d) simulations illustratingthe evolution from unstable in-phase to stable anti-phase oscillations Damped beating is foundin both the pendulumsrsquo (a) (c) phase direg erence not and (b) (d) amplitudes A1 A2 Time isscaled by the pendulum period and amplitudes are scaled by the initial amplitude at t = 0 Theexperimental conditions correspond to those listed in macrgure 1

Though these aspects of energy damping and energy input were not included in hisquantitative analysis Korteweg concluded that Huygensrsquos observations were entirelycaptured by the three-degree-of-freedom model and that the anti-phase mode if notthe only sustainable motion enjoyed a distinct advantage over in-phase motion Thisadvantage was cleverly used in a dinoterent context for precise measurements of theacceleration of gravity by Vening Meinesz and others who used pairs of pendulumsset into free (no driving) anti-phase oscillation (Heiskanen amp Vening Meinesz 1958)

Blekhman (1988) also discusses Huygensrsquos observations in his book and he re-counts the results of a laboratory reproduction of the coupled clocks as well as pre-senting a theoretical analysis of oscillators coupled through a common supportingframe His model is similar to the one studied by Korteweg except that Blekhmanuses van der Pol oscillators rather than pendulums so that his quantitative analysisincludes both (weak) driving and damping enotects Blekhman was primarily moti-vated by the general aspects of synchronization phenomena found in a wide vari-ety of physical systems and it is probably for this reason that he chose to usevan der Pol oscillators He predicted that both in-phase and anti-phase motionsare stable under the same circumstances (that is the two are coexisting attrac-tors) Somewhat puzzling is that he reports observing both states in the exper-imental reproduction while this agrees with his predictions for the van der Polsystem Huygens (so far as we know) never mentioned stable in-phase synchroniza-tion

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 565

2 Background

(a) Details of Huygensrsquos clock experiments

The operation of a pendulum clock is described in great detail by Huygens (Huygens1986) In brief the pendulum is attached to an escapement which alternately blocksand releases a scape wheel as the pendulum oscillates This action provides the timingthat is transmitted from the scape wheel to the clocksrsquo hands via a gearing system(the motion work ) The scape wheel is also connected through a separate gearingsystem (the going train) to an energy source typically a wound spring or elevatedweights The unwinding spring or falling weights drive the motion of the scape wheelwhich in turn provides small `kicksrsquo to the pendulum via the escapement Thistransmission of energy to the pendulum compensates for losses due to friction thusthe pendulum continues to oscillate indeshy nitely as long as the spring is periodicallyrewound or the fallen weights are periodically raised back up

Many experimental details of Huygensrsquos observations are recorded in his writings(Huygens 1950a b Huygens 1986) The pendulum in each clock measured ca 9 iny inlength corresponding to an oscillation period of ca 1 s Each pendulum weighed 1

2lbz

and regulated the clock through a verge escapement which required each pendulumto execute large angular displacement amplitudes of ca 20macr or more from verticalfor the clock to function (see Rawlings (1944) and also Landes (1983 appendix A)for a detailed description of the verge escapement) The amplitude dependence ofthe period in these clocks was typically corrected by use of cycloidal-shaped bound-aries to conshy ne the suspension (Huygens 1986) Each pendulum clock was enclosedin a 4 ft long case a weight of ca 100 lb was placed at the bottom of each case(to keep the clock oriented aboard a ship) From this information we estimate thatthe important experimental parameter middot the ratio of the single pendulum mass tothe total system mass is ca 0005 Though the two clocks dinotered in certain phys-ical aspects (for example Huygens notes that the size of the clocks was somewhatdinoterent (Huygens 1950a b)) the clocks were closely matched in those characteris-tics we expect are essential for the dynamics In particular well-adjusted pendulumclocks of the 1660s would typically run at rates which dinotered by only 15 s per day(so their natural frequencies dinotered by approximately two parts in 10 000) (Landes1983) Pendulum clocks represented a tremendous advance in horology the science oftimekeeping Before Huygensrsquos invention of the pendulum clock typical clocks (egverge escapement with balance wheel regulator) varied by ca 15 min (1) per day(Landes 1983)

Huygensrsquos laboratory notebook contains a detailed description of tests and observa-tions on synchronization (Huygens 1950a b) In some experiments Huygens studiedtwo clocks that were suspended side by side each hanging from a hook embeddedin the same wooden beam In other experiments Huygens studied a conshy gurationwith each clock hanging from its own wooden beam and the two beams lying ontop of back to back chairs At shy rst Huygens suspected the `sympathyrsquo was dueto induced air currents but eventually concluded the cause was the `imperceptiblemovementsrsquo of the common supporting structure The associated coupling was weakwhen Huygens disturbed the pendulums he found the clocks required ca 30 min

y 1 in equals 254 cmz 1 lb equals 04536 kg 1 ft equals 12 in equals 305 cm

Proc R Soc Lond A (2002)

566 M Bennett and others

(a)

(b)

(1) (2)

Figure 1 Multiple-exposure images illustrate the anti-phase attractor of a large mass ratio( middot = 00063) system of two pendulums mounted on a common translating beam (a) The systemis started with equal amplitude in-phase oscillation of the left (1) and the right (2) pendulumsIn the presence of weak coupling through lateral motion of the mounting cart the in-phase stateis unstable and after a suplusmn cient time the system oscillates stably in anti-phase macrnal state (b)

before synchronization was restored At one second per cycle a transient time of30 min amounts to 1800 cycles (Interestingly Huygens considers this restorationtime as fast perhaps a reregection of the pace of life then and now) The dissipationwas also weak as can be seen from the following estimate the total energy avail-able to run each clock is ca 100 J obtained from weights of ca 10 kg mass fallinga distance of 1 m Assuming this energy is sumacr cient to keep a clock running forca 250 000 oscillations (ca 3 days) the energy input per oscillation is ca 4 pound 10iexcl4 JThe total energy in a 9 in long 1

2lb pendulum swinging through a semiarc of 25macr

is ca 005 J Thus energy loss per oscillation relative to the pendulum energy isca 08

(b) Recapping Korteweg and Blekhman studies

We know of two studies which were directly motivated by Huygensrsquos observationsThe shy rst was Kortewegrsquos (1906) paper in which he analysed a three-degree-of-freedommodel consisting of two plane pendulums connected to a rigid frame free to oscillatein one dimension (Korteweg was also strongly motivated by the 18th century obser-vations of Ellicott regarding the slow beating between weakly coupled pendulums)Korteweg made a linear normal mode analysis for small oscillations in the absenceof damping and driving enotects To explain why only certain of these modes might beobserved and others not Korteweg introduced the idea that friction is responsiblefor certain motions being unsustainable If a mode involved large-amplitude motionof the supporting frame he argued the internal clock mechanisms would be unableto provide enough energy to sustain this motion Conversely for a mode in whichthe frame moved only a little the energy input could overcome the enotects of friction

Proc R Soc Lond A (2002)

Huygensrsquos clocks 567

0 1000089

094

04

07

10

0

p 2

p

t

A

A1

A2

a(r

ad)

0 1000t

(a) (b)

(c) (d)

Figure 2 (a) (b) Time-series from pendulum experiments and (c) (d) simulations illustratingthe evolution from unstable in-phase to stable anti-phase oscillations Damped beating is foundin both the pendulumsrsquo (a) (c) phase direg erence not and (b) (d) amplitudes A1 A2 Time isscaled by the pendulum period and amplitudes are scaled by the initial amplitude at t = 0 Theexperimental conditions correspond to those listed in macrgure 1

Though these aspects of energy damping and energy input were not included in hisquantitative analysis Korteweg concluded that Huygensrsquos observations were entirelycaptured by the three-degree-of-freedom model and that the anti-phase mode if notthe only sustainable motion enjoyed a distinct advantage over in-phase motion Thisadvantage was cleverly used in a dinoterent context for precise measurements of theacceleration of gravity by Vening Meinesz and others who used pairs of pendulumsset into free (no driving) anti-phase oscillation (Heiskanen amp Vening Meinesz 1958)

Blekhman (1988) also discusses Huygensrsquos observations in his book and he re-counts the results of a laboratory reproduction of the coupled clocks as well as pre-senting a theoretical analysis of oscillators coupled through a common supportingframe His model is similar to the one studied by Korteweg except that Blekhmanuses van der Pol oscillators rather than pendulums so that his quantitative analysisincludes both (weak) driving and damping enotects Blekhman was primarily moti-vated by the general aspects of synchronization phenomena found in a wide vari-ety of physical systems and it is probably for this reason that he chose to usevan der Pol oscillators He predicted that both in-phase and anti-phase motionsare stable under the same circumstances (that is the two are coexisting attrac-tors) Somewhat puzzling is that he reports observing both states in the exper-imental reproduction while this agrees with his predictions for the van der Polsystem Huygens (so far as we know) never mentioned stable in-phase synchroniza-tion

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

566 M Bennett and others

(a)

(b)

(1) (2)

Figure 1 Multiple-exposure images illustrate the anti-phase attractor of a large mass ratio( middot = 00063) system of two pendulums mounted on a common translating beam (a) The systemis started with equal amplitude in-phase oscillation of the left (1) and the right (2) pendulumsIn the presence of weak coupling through lateral motion of the mounting cart the in-phase stateis unstable and after a suplusmn cient time the system oscillates stably in anti-phase macrnal state (b)

before synchronization was restored At one second per cycle a transient time of30 min amounts to 1800 cycles (Interestingly Huygens considers this restorationtime as fast perhaps a reregection of the pace of life then and now) The dissipationwas also weak as can be seen from the following estimate the total energy avail-able to run each clock is ca 100 J obtained from weights of ca 10 kg mass fallinga distance of 1 m Assuming this energy is sumacr cient to keep a clock running forca 250 000 oscillations (ca 3 days) the energy input per oscillation is ca 4 pound 10iexcl4 JThe total energy in a 9 in long 1

2lb pendulum swinging through a semiarc of 25macr

is ca 005 J Thus energy loss per oscillation relative to the pendulum energy isca 08

(b) Recapping Korteweg and Blekhman studies

We know of two studies which were directly motivated by Huygensrsquos observationsThe shy rst was Kortewegrsquos (1906) paper in which he analysed a three-degree-of-freedommodel consisting of two plane pendulums connected to a rigid frame free to oscillatein one dimension (Korteweg was also strongly motivated by the 18th century obser-vations of Ellicott regarding the slow beating between weakly coupled pendulums)Korteweg made a linear normal mode analysis for small oscillations in the absenceof damping and driving enotects To explain why only certain of these modes might beobserved and others not Korteweg introduced the idea that friction is responsiblefor certain motions being unsustainable If a mode involved large-amplitude motionof the supporting frame he argued the internal clock mechanisms would be unableto provide enough energy to sustain this motion Conversely for a mode in whichthe frame moved only a little the energy input could overcome the enotects of friction

Proc R Soc Lond A (2002)

Huygensrsquos clocks 567

0 1000089

094

04

07

10

0

p 2

p

t

A

A1

A2

a(r

ad)

0 1000t

(a) (b)

(c) (d)

Figure 2 (a) (b) Time-series from pendulum experiments and (c) (d) simulations illustratingthe evolution from unstable in-phase to stable anti-phase oscillations Damped beating is foundin both the pendulumsrsquo (a) (c) phase direg erence not and (b) (d) amplitudes A1 A2 Time isscaled by the pendulum period and amplitudes are scaled by the initial amplitude at t = 0 Theexperimental conditions correspond to those listed in macrgure 1

Though these aspects of energy damping and energy input were not included in hisquantitative analysis Korteweg concluded that Huygensrsquos observations were entirelycaptured by the three-degree-of-freedom model and that the anti-phase mode if notthe only sustainable motion enjoyed a distinct advantage over in-phase motion Thisadvantage was cleverly used in a dinoterent context for precise measurements of theacceleration of gravity by Vening Meinesz and others who used pairs of pendulumsset into free (no driving) anti-phase oscillation (Heiskanen amp Vening Meinesz 1958)

Blekhman (1988) also discusses Huygensrsquos observations in his book and he re-counts the results of a laboratory reproduction of the coupled clocks as well as pre-senting a theoretical analysis of oscillators coupled through a common supportingframe His model is similar to the one studied by Korteweg except that Blekhmanuses van der Pol oscillators rather than pendulums so that his quantitative analysisincludes both (weak) driving and damping enotects Blekhman was primarily moti-vated by the general aspects of synchronization phenomena found in a wide vari-ety of physical systems and it is probably for this reason that he chose to usevan der Pol oscillators He predicted that both in-phase and anti-phase motionsare stable under the same circumstances (that is the two are coexisting attrac-tors) Somewhat puzzling is that he reports observing both states in the exper-imental reproduction while this agrees with his predictions for the van der Polsystem Huygens (so far as we know) never mentioned stable in-phase synchroniza-tion

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 567

0 1000089

094

04

07

10

0

p 2

p

t

A

A1

A2

a(r

ad)

0 1000t

(a) (b)

(c) (d)

Figure 2 (a) (b) Time-series from pendulum experiments and (c) (d) simulations illustratingthe evolution from unstable in-phase to stable anti-phase oscillations Damped beating is foundin both the pendulumsrsquo (a) (c) phase direg erence not and (b) (d) amplitudes A1 A2 Time isscaled by the pendulum period and amplitudes are scaled by the initial amplitude at t = 0 Theexperimental conditions correspond to those listed in macrgure 1

Though these aspects of energy damping and energy input were not included in hisquantitative analysis Korteweg concluded that Huygensrsquos observations were entirelycaptured by the three-degree-of-freedom model and that the anti-phase mode if notthe only sustainable motion enjoyed a distinct advantage over in-phase motion Thisadvantage was cleverly used in a dinoterent context for precise measurements of theacceleration of gravity by Vening Meinesz and others who used pairs of pendulumsset into free (no driving) anti-phase oscillation (Heiskanen amp Vening Meinesz 1958)

Blekhman (1988) also discusses Huygensrsquos observations in his book and he re-counts the results of a laboratory reproduction of the coupled clocks as well as pre-senting a theoretical analysis of oscillators coupled through a common supportingframe His model is similar to the one studied by Korteweg except that Blekhmanuses van der Pol oscillators rather than pendulums so that his quantitative analysisincludes both (weak) driving and damping enotects Blekhman was primarily moti-vated by the general aspects of synchronization phenomena found in a wide vari-ety of physical systems and it is probably for this reason that he chose to usevan der Pol oscillators He predicted that both in-phase and anti-phase motionsare stable under the same circumstances (that is the two are coexisting attrac-tors) Somewhat puzzling is that he reports observing both states in the exper-imental reproduction while this agrees with his predictions for the van der Polsystem Huygens (so far as we know) never mentioned stable in-phase synchroniza-tion

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

568 M Bennett and others

3 Experimental realization and results

We re-examine Huygensrsquos synchronization observations in an experiment with twopendulum clocks mounted side by side on a single wooden beam (shy gure 1) Thependulum clocks are commercially available spring-wound time pieces (Model 771-000 Uhrenfabrik Franz Hermle amp Sohn Gosheim Germany) Each clock contains a140 cm pendulum (with a nominal frequency of 133 Hz) of mass m = 0082 kg thependulum is coupled to an anchor escapement which enables the clock movementto function with small angular displacements of ca 8macr from vertical The beam ismounted on a low-friction wheeled cart (Model ME-9454 Pasco Scientishy c RosevilleCA) The combined system of clocks beam and cart is placed atop a slotted track(ME-9429A Pasco Scientishy c) which permits the system to translate freely in adirection parallel to the beam The total mass of the cart and clocks without thependulums is M Weights are added to and removed from the cart to change Mand thereby to change the system mass ratio middot sup2 m=(2m + M ) The motion ofeach pendulum is monitored by tracking a laser beam reregected from the pendu-lum suspension using a position-sensing detector (Model 1L30 On-Trak PhotonicsLake Forest CA) The lasers and detectors (not shown in shy gure 1) are mountedon the system permitting measurement of each pendulumrsquos angular position in thesystem reference frame The voltage signal from each detector is recorded using acomputer-based data-acquisition system complex demodulation of the signals yieldsmeasurement of each pendulumrsquos oscillation amplitude frequency and phase as afunction of time

The clocks synchronize in anti-phase when the system mass ratio middot is compara-ble with that reported by Huygens (shy gures 1 and 2) In this case the anti-phasestate is the attractor when the system begins from any `goodrsquo initial conditionwhich ensures that each clock is initially functioning (Starting one pendulum atrest with zero angular displacement is an example of a `badrsquo initial condition theenergy exchange between the coupled pendulums is not sumacr cient to jump-start apendulum whose initial amplitude is too small to engage the escapement) Con-sider for example the case where the system starts at rest with both pendulumshaving equal amplitude in-phase angular displacements (shy gure 1) The approach tothe anti-phase state is slow occurring over the course of several hundred pendulumoscillations (shy gure 2a) The phase dinoterence not = not 1 not 2 exhibits some overshootand small slow variations in not about ordm persist indeshy nitely The complex demod-ulated pendulum amplitudes A1 and A2 initially exhibit slow approximately out-of-phase oscillations in a manner characteristic of beating between weakly coupledlinear oscillators (shy gure 2b) These beating oscillations are damped and eventuallythe amplitudes become nearly steady and approximately equal as not gets close toordm During this evolution the amplitude of the cartrsquos motion is typically very small(ca 01 mm)

Stable anti-phase synchronization requires the pendulum clocks to be very closelymatched in frequency For example anti-phase synchronization is observed with middot =00063 when the dinoterence between the natural frequencies of the clocks is 00009 HzBy simply exchanging the pendulum bobs between these two (very similar but notidentical) clocks this frequency dinoterence is increased to 00045 Hz and anti-phasesynchronization no longer occurs Instead the two clocks run `uncoupledrsquo at theirindividual frequencies

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 569

10

05

0

0007 0010 0013micro

anti

phas

e pr

obab

ilit

y

decreasing mass ratioincreasing mass ratio

Figure 3 Probability of observing stable anti-phase oscillations as a function of mass ratio middot forthe clock system that begins at rest with both pendulums having in-phase angular displacementsof equal amplitude In this range of middot only two asymptotic states are observed anti-phaseoscillation and beating death

When middot is sumacr ciently increased we shy nd that some initial conditions can leadto a state which we call `beating deathrsquo where one or both of the clocks cease tofunction (shy gure 3) If the angular displacement of either pendulum clock falls below aminimum threshold the escapement mechanism can no longer engage the pendulumseizes and the clock stops When the pendulums start from rest with in-phase angulardisplacements of equal amplitude anti-phase oscillations are always observed formiddot lt 00083 while the clocks typically stop for middot gt 00125 At intermediate values ofmiddot either state may be the attractor of the system depending on slight dinoterences inthe initial conditions Additionally as a function of middot the system exhibits hysteresiswhich may be due in part to the dependence of the track friction on the mass loadingof the system

We believe that our clock system contains the same essential ingredients as Huy-gensrsquos clock system In both cases the clocks are mounted on a common supportwhose motion provides weak coupling Dissipation is also weak both in Huygensrsquosclocks and ours in our case we estimate the relative energy loss per oscillation isca 3 Both systems of clocks are kept out of equilibrium by the inherently nonlineardriving from small impulsive kicks applied by the escapement mechanisms when thependulumsrsquo displacement amplitudes exceed a threshold value Of course Huygensrsquosclocks dinoter in certain details that are qualitatively unimportant as follows

(a) Huygensrsquos clocks were driven by falling weights ours are spring driven

(b) Huygensrsquos clocks used a verge escapement which required large displacementsamplitudes of ca 25macr to function our clocks use a newer (invented in the 1670s)anchor design which enables clocks to function with smaller amplitudes

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

570 M Bennett and others

(c) The length of Huygensrsquos oscillating pendulums was continually varied by con-shy ning the suspension of each pendulum between cycloidal-shaped boundariesour pendulums oscillated with a shy xed suspension length

4 Theory model and analysis

We consider the three-degree-of-freedom model depicted in shy gure 4 Two plane pen-dulums hang from a common rigid frame which is constrained to move in one dimen-sion The pendulums are identical each consisting of a point mass hanging from amassless rigid rod In the absence of damping and driving the Lagrangian is

L = 12(M + 2m) _X2 + m _X`(cos iquest 1

_iquest 1 + cos iquest 2_iquest 2) + 1

2m`2( _iquest 2

1 + _iquest 22)

+ mg`(cos iquest 1 + cos iquest 2) 12KX2 (41)

where iquest k is the angular displacement of the kth pendulum about its pivot pointX is the linear displacement of the platform m is the pendulum mass M is theplatform mass g is the acceleration due to gravity ` is the pendulum length andthe overdot denotes dinoterentiation with respect to time We take the platform motionto be weakly bound by a harmonic restoring force since in Huygensrsquos experimentsthe common supporting beam was conshy ned In our experiments the platform is freeto slide so that K = 0 (In earlier experiments we included a conshy ning force bymounting magnets on the platform base The results were similar and for practicalsimplicity we shy nally settled on the K = 0 design) In the analysis to follow we keepthe harmonic restoring force to handle both cases and to check that this dinoterenceis indeed a minor detail

We also add viscous damping to the pendulums and the platform and a drivingmechanism to model the clocks escapements The governing equations of motionbecome

iquest k + b _iquest k +g

`sin iquest k =

1

`X cos iquest k + ~fk (42)

(M + 2m) X + B _X + KX =X

j

m`(sin iquest k) (43)

where b and B are friction coemacr cients The clock mechanism represented by ~fkprovides the energy needed to keep the clock running (see below)

It is convenient to write the dinoterential equations in dimensionless form introduc-ing a scaled position Y = X=` and time frac12 = t

pg=` so that

iquest 00k + 2 reg iquest 0

k + sin iquest k = Y 00 cos iquest k + fk (44)

Y 00 + 2iexcl Y 0 + laquo 2Y = middot (sin iquest 1 + sin iquest 2)00 (45)

where reg = bp

`=4g iexcl = Bp

`=4g=(M + 2m) laquo 2 = K=(M + 2m) middot = m=(M + 2m)and the prime denotes dinoterentiation with respect to frac12 The system mass ratio middot =m=(M + 2m) controls the coupling strength and is a key parameter in our analysis

Rather than develop a detailed model of the escapement mechanism (Lepschy etal 1992) we use a simple impulse rule when the pendulum reaches a threshold anglesect copy the angular velocity reverses direction and its magnitude changes according to

j iquest 0k j (1 c)j iquest 0

k j + deg (46)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 571

X

f 1 f 2

Figure 4 Sketch of model coupled-pendulum system

where c and deg are small positive constants Our choice of impulse rule is loosely basedon the two-part action of the escapement which shy rst engages at a shy xed angle andthen delivers a kick on the downswing as the anchor hits Since we are interested inthe heavy platform limit M frac34 m we ignore the reaction force on the platform

In what follows we consider small-angle swings only Even so the problem isnonlinear due to the impulsive kicks Between kicks the dynamics is linear howeverand the motion can be decomposed into a superposition of independent normal modeoscillations We exploit this below in deriving a Poincare map alternately applyingthe normal mode evolution and the instantaneous enotect of the kicks

Before turning to a derivation of the iterative map we pause to present a simpleexplanation of Huygensrsquos observation Introducing sum and dinoterence variables frac14 =iquest 1 + iquest 2 and macr = iquest 1 iquest 2 equations (44) and (45) become for small oscillations andbetween kicks

macr 00 + 2 reg macr 0 + macr = 0 (47)

frac14 00 + 2 reg frac14 0 + frac14 = 2Y 00 (48)

Y 00 + 2 iexcl Y 0 + laquo 2Y = middot frac14 00 (49)

Only the sum coordinate couples to the platform motion Thus the damping in theplatform anotects frac14 but not macr To take the most extreme situation if the pendulumswere free of friction (ie reg = 0) the coordinate frac14 would still damp out only macrsurvives and this corresponds to pure anti-phase motion just as Huygens observed

This argument is instructive and has the essential ingredients but is incompletesince it ignores the energy input altogether Without the kicks the amplitude of thesurviving (anti-phase) oscillations would depend on the initial conditions (contraryto observations) and for nearly in-phase initial conditions the anti-phase amplitudewould be so small that the clock escapement would not engage so that the anti-phasestate would be unsustainable even in principle

Nevertheless the normal modes play a central role in the analysis that followsUsing the (complex) notation macr (t) = macr jeij t frac14 (t) = frac14 jeij t and Y (t) = Yjeij t todeshy ne the jth mode we can determine the three mode frequencies j and the corre-sponding mode coordinates ordfj One mode follows from the decoupling of the dinoter-ence coordinate

1 = 1 + i reg ordf1 = ( macr frac14 Y ) = (1 0 0) (410)

This describes pure anti-phase motion with decay rate Im(1) = reg Here and inwhat follows we will assume for convenience that the damping is weak ( reg frac12 1)and so neglect terms of order reg 2 The exact expressions for the other two modes arecumbersome but for our purposes it is enough to develop them as a power series in

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

572 M Bennett and others

the mass ratio middot We shy nd

2 = 1 + i reg + middot (1 + 2i( reg + iexcl )) ordf2 = (0 1 middot d[1 2id( iexcl laquo 2 reg )]) (411)

and

3 = laquo + i iexcl + O( middot ) ordf3 = (0 2 laquo 2 1) + O( middot ) (412)

where diexcl1 = laquo 2 1 and we have written these in a form suitable for the case wherethe platform is underdamped ( laquo gt iexcl ) In fact this is not a necessary assumptionWe only want to avoid the resonant case laquo = 1 when the platform frequency matchesthe pendulum frequency so we assume either the frequencies do not match or theplatform motion is overdamped

Consideration of the mode structure allows us to simplify our analysis First sincewe want to consider the situation where the platform has `imperceptible movementsrsquoY is small But the only linear combinations of the three modes

Pj cjordfj which yield

a small value for Y have an equally small amount of the third mode c3=c2 sup1 O( middot )This means that the third mode is barely excited Second the remaining modes ordf 1

and ordf 2 are only O( middot ) dinoterent from macr and frac14 respectively Thus if a kick from theescapement mechanism boosts the value of iquest 1 say then to leading order in middot itanotects modes ordf 1 and ordf 2 and not ordf 3

Together these observations allow us to ignore the third mode thereby reducingthe problem to two degrees of freedom We now proceed to construct a Poincare mapin the reduced four-dimensional phase space which is spanned by the coordinates( iquest 1 iquest 0

1 iquest 2 iquest 02) Our strategy is to exploit the weakness of the damping and coupling

During one oscillation we imagine that each pendulum executes a nearly free har-monic orbit and then compute the small changes due to the damping and drivingThus we deshy ne

iquest j = Aj sin( frac12 + not j) j = 1 2 (413)

where the iterative map describes updates in the amplitudes Aj and phases not j Inthe phase plane for each pendulum one can picture the free orbit as uniform motionaround the circle of radius Aj except that when it reaches the positions iquest j = copywhen iquest 0

j gt 0 and iquest j = copy when iquest 0j lt 0 the pendulum suddenly changes sign due

to the kick from the clock mechanism (see shy gure 5) During the remaining motionthere is energy loss due to friction

The map we will derive considers the moment when the shy rst pendulum passesthrough its lowest point moving to the right so the Poincare section is iquest 1 = 0iquest 0

1 gt 0 We consider in turn the enotect due to (1) the damping and (2) the fourimpacts with the clock mechanisms (two impacts per pendulum per cycle)

(a) Damping

The enotect of damping is most easily expressed by using the normal mode co-ordinates which are to leading order just the sum and dinoterence combinations frac14and macr Introducing the `polarrsquo representation corresponding to equation (413) wecan write

frac14 = A+ sin( frac12 + not + ) (414)

macr = Aiexcl sin( frac12 + not iexcl) (415)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 573

where

A2sect = A2

1 + A22 sect 2A1A2 cos( not 1 not 2) (416)

tan not sect =A1 sin not 1 sect A2 sin not 2

A1 cos not 1 sect A2 cos not 2 (417)

and so

4A2j = A2

+ + A2iexcl sect 2A+ Aiexcl cos( not + not iexcl) (418)

tan not j =A+ sin not + sect Aiexcl sin not iexcl

A+ cos not + sect Aiexcl cos not iexcl (419)

where the upper signs correspond to j = 1 and the lower signs to j = 2The normal modes evolve independently with the real and imaginary parts of

each mode frequency determining the time variation of the mode amplitude andphase respectively If we denote the mode frequencies by + and iexcl for in-phaseand anti-phase respectively over one oscillation we have A + sup1 A + Aiexcl cedil Aiexcl and (not + not iexcl ) ( not + not iexcl ) + shy where the constants sup1 cedil and shy are

sup1 = eiexcl2ordm Im + (420)

cedil = eiexcl2ordm Im 1 (421)

shy = 2 ordm Re( + iexcl ) (422)

One can work out what this corresponds to in terms of the variables (A1 A2 not 1 not 2)using equations (418) and (419) Denoting the new values by a tilde the result is

4 ~A21 = ( sup1 + cedil )2A2

1 + ( sup1 cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not + shy 4 sup1 cedil A1A2 sin not (423)

4 ~A22 = ( sup1 cedil )2A2

1 + ( sup1 + cedil )2A22 + 2( sup1 2 cedil 2)A1A2 cos not shy 4 sup1 cedil A1A2 sin not (424)

tan ~not =2 sup1 cedil [2A1A2 sin not + shy (A2

1 A22)]

( sup1 2 cedil 2)(A21 + A2

2) + 2( sup1 2 + cedil 2)A1A2 cos not (425)

wherenot = not 1 not 2 (426)

and we have kept terms to shy rst order in shy Physically shy is the coupling-inducedfrequency shift between the otherwise degenerate normal modes which of course issmall for small coupling Subtracting the old values from the new ones we get

cent dAj = ~Aj Aj (427)

cent d not = ~not not (428)

where the notation cent d indicates the changes due to damping in the amplitudes andthe phase dinoterences

(b) Impacts

Since the impacts `truncatersquo the free orbit near the turning points the pendulumperiods are less than 2 ordm (shy gure 5) This amounts to advancing the phases not j Thesituation is particularly simple if considered in terms of the single-oscillator phasespace the time taken to traverse the truncated orbit is directly proportional to the

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

574 M Bennett and others

f

f rsquo

a j Aj

Figure 5 Phase portrait of single freersquo orbit truncated at iquest j = sect copy

perimeter of the truncated circle so that it takes a time T = 4 arcsin( copy =Aj) for thejth pendulum to return to the section With cent i indicating the changes due to theimpacts the change in not j is

cent i not j = 2ordm 4 arcsin( copy =Aj) j = 1 2 (429)

so thatcent i not = cent i not 1 cent i not 2 (430)

The enotect on the amplitudes can be calculated by energy considerations Notingthat iquest 2

j + iquest 02j = A2

j is constant over the unperturbed orbit the angular speed atiquest j = sect copy is

j iquest 0j j =

qA2

j copy 2 (431)

Applying the impulse rule equation (46) twice

j iquest 0j j (1 c)[(1 c)j iquest 0

j j + deg ] + deg (432)

which gives the angular speed the instant after the second kick Using equation (431)again gives the new amplitude and thus the change due to the impacts

cent iAj =n

copy 2 + [(1 c)2q

A2j copy 2 + (2 c) deg ]2

o1=2

Aj (433)

These changes in phase and amplitude apply only if the amplitude is big enoughto trigger the escapement ie only if Aj gt copy

(c) Analysis of the map

Together the contributions from the impacts and the damping give a three-dimensional return map for the coordinates (A1 A2 not )

A1 A1 + cent iA1 + cent dA1 (434)

A2 A2 + cent iA2 + cent dA2 (435)

not not + cent i not + cent d not (436)

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 575

We can identify three shy xed points all of which have equal amplitudes (A1 = A2)One is the trivial solution A1 = A2 = 0 To shy nd the other two note shy rst that if A1 =A2 then cent i not 1 = cent i not 2 from equation (429) and cent iA1 = cent iA2 from equation (433)Meanwhile in equations (423) and (424) setting not = 0 or ordm and A1 = A2 implies~A1 = ~A2 while taking A1 = A2 and the limit not 0 or ordm in equation (425) yields~not 0 or ordm respectively Thus we have two non-trivial shy xed points one with not = 0and the other with not = ordm These are the in-phase and anti-phase states respectively

Before we discuss the behaviour of the full three-dimensional map it is useful toconsider the special case A1 = A2 gt copy and shy = 0 We can get a fairly completepicture in this case The subspace A1 = A2 is invariant and contains the shy xedpoints identishy ed above Moreover the phase dinoterence not obeys its own map Fromequation (429) we see that not is unchanged due to the impulsive kicks and fromequation (425) that the amplitudes cancel out leading us to the one-dimensionalmap

~not = arctan

frac122 sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (437)

For any sup1 cedil there are exactly two shy xed points not = 0 and ordm These are just thenon-trivial shy xed points already identishy ed By looking at the slope of this map onereadily shows that not = 0 is locally unstable and not = ordm is locally stable for all allowedsup1 cedil

There is an alternative way to analyse the equal-amplitude case which allows us tomake a stronger statement about the stability of the anti-phase state The argumentfocuses directly on the mode energies which gives it a certain conceptual advan-tage The mode energies are proportional to the squared amplitudes A2

sect SettingA1 = A2 = A in equation (416) yields

A2sect = 2A2(1 sect cos( not 1 not 2)) (438)

Now we have established (cf equation (429)) that if A1 = A2 the impulses do notanotect not Thus if the impulse causes an amplitude change A A + cent A we havefrom equation (438)

Asect A + cent A

AAsect

which says that A+ and Aiexcl are scaled by the same factor Meanwhile frictionaldamping also acts to scale A + and Aiexcl

A + sup1 A + Aiexcl cedil Aiexcl

where 0 6 sup1 lt cedil 6 1 Taken together the two contributions yield the map

~A + = sup1

micro1 +

cent A

A

paraA + (439)

~Aiexcl = cedil

micro1 +

cent A

A

paraAiexcl (440)

Note that this does not give an explicit description of the systemrsquos evolution since Aitself is a dynamical variable Nevertheless it is enough to draw the strong conclusionthat the anti-phase state is globally attracting (within the equal amplitude subspace)This follows because since sup1 lt cedil proportionately more energy is drained out of the

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

576 M Bennett and others

in-phase mode than is put into it in all cases except if the system is perfectly in-phase(and so Aiexcl = 0)

Let us return now to consider the full three-dimensional map Numerical simula-tions of the three-dimensional map reveal that the anti-phase state and the trivialstate are asymptotically stable while the in-phase state is unstable consistent withHuygensrsquos observations (and ours) As already noted shy is the frequency dinoterencebetween the two normal modes when shy 6= 0 the main enotect is the introduction ofbeats which eventually damp out (see shy gure 2) Because of these even if the pendu-lums initially have the same amplitude this property does not persist Neverthelessthe beating merely introduces a transient which serves to decorate the main trendsin the systemrsquos evolution ie those trends which we previously deduced by settingshy = 0 without anotecting the long-term behaviour

The map behaves very similarly to our experiments and Huygensrsquos observationsFigure 2 shows a typical situation even when started close to the in-phase state thesystem evolves into the anti-phase state with damped beating behaviour in boththe amplitudes and the phase dinoterence More generally the model qualitativelyreproduces the important features of the experimental data though there are somedinoterences The chief discrepancy is that the model shows a much smaller modulationof the amplitudes a property that can be traced to the sharp impulse mechanism(see equation (46))

This is not the end of the story however Simulations reveal another type ofattracting state where one pendulum oscillates but the other does not we call thisbehaviour `beating deathrsquo because beating plays an important role in determiningthe shy nal dynamical state Beating death happens when the pendulum amplitudesget very close to the escapement threshold copy If both pendulums have sub-thresholdamplitudes there is no energy input and the system is attracted to the trivial stateA1 = A2 = 0 However if the two amplitudes are slightly larger than copy the beatingcan introduce a large enough dinoterence between the two so that the motion of onependulum dies out while the other does not This type of shy nal state is also observedin our experiments when the coupling is sumacr ciently strong

Drawing everything so far together we shy nd that depending on initial conditions thesystem ends up either in the anti-phase state or in beating death The latter occursmore often the greater the coupling all other parameters held shy xed We can estimatewhen we expect to observe beating death with signishy cant probability by consideringthe in-phase solution (for which damping enotects are greatest) with A1 = A2 andcomputing the value of the common value A when this falls below copy the motioncannot be sustained This leads to the following condition for beating death

1

2 copy(2 c)2 deg 2 2 ordm reg 4 ordm middot ( reg + iexcl ) lt 0 (441)

The shy rst term is the amplitude boost due to the clock mechanism (when A = copy )the rest is the amplitude loss due to friction in pendulum and platform It is the middot -dependent term that is interesting from the point of view of Huygensrsquos observationssince presumably his clocks were run under the condition that in isolation eachclock maintains its oscillations We see plainly that for large enough middot the conditionfor quiescence is satisshy ed

Finally we consider the enotect of non-identical clocks With identical pendulumsthe anti-phase state is attracting for arbitrarily small values of middot However in general

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

Huygensrsquos clocks 577

m

D

(a)

m

G

(b)

quasiperiodicbeating death

antiphasequasiperiodic

antiphase

beating death

experimental path

Figure 6 Phase diagrams summarizing theoretical analysis(a) cent middot parameter plane (b) iexcl middot parameter plane

one expects that two oscillators with dinoterent natural frequencies will not frequencylock unless the coupling exceeds some minimum threshold value We can get an esti-mate of this threshold by the following argument Assuming that the amplitudesare very nearly equal and that the beating parameter shy is sumacr ciently small thatthey stay nearly equal the relative phase evolves according to the map (see equa-tion (437))

~not = cent + arctan

frac122sup1 cedil sin not

( sup1 2 cedil 2) + ( sup1 2 + cedil 2) cos not

frac34 (442)

where a constant cent has been added to account for the dinoterence in pendulum fre-quencies (More specishy cally if these frequencies dinoter by an amount raquo then in theuncoupled case not advances by an amount raquo T where T is the mean period) As thedetuning is increased the not = 0 shy xed point shifts up and the not = ordm shy xed point shiftsdown until they coincide at the critical detuning cent c beyond which the pendulumsare not frequency locked Since both sup1 and cedil dinoter only a little from unity we canshy nd cent c by setting not = ordm =2 in equation (442) with the result

cent c ordm sup1 cedil

cedil (443)

For example taking laquo 2 frac12 1 this becomes

cent c = 4 ordm middot ( iexcl + reg ) (444)

5 Discussion and serendipity

Figure 6 summarizes our theoretical results The shy gure depicts which of the threetypes of attracting states predominate as a function of system parameters The statelabelled `quasiperiodicrsquo refers to the case where the pendulums run at dinoterent fre-quencies Near each boundary the system can end up in one or other state dependingon initial conditions We have made two plots in order to piece together the full pic-ture as a function of the three key parameters In shy gure 6a we hold the platformdamping iexcl shy xed and vary the detuning cent and coupling strength middot The anti-phase

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

Birch T 1756 The history of The Royal Society of London for improving of natural knowledge inwhich the most considerable of those papers communicated to the Society which have hithertonot been published are inserted in their proper order as a supplement to the PhilosophicalTransactions vol 2 pp 19 21 2324 London Johnson (Reprint 1968)

Blekhman I I 1988 Synchronization in science and technology New York ASME

Britten F J 1973 Brittenrsquos old clocks and watches and their makers a historical and descrip-tive account of the direg erent styles of clocks and watches of the past in England and abroadcontaining a list of nearly fourteen thousand makers London Methuen

Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

Heiskanen W A amp Vening Meinesz F A 1958 The Earth and its gravity macreld McGraw-Hill

Huygens C 1669 Instructions concerning the use of pendulum-watches for macrnding the longitudeat sea Phil Trans R Soc Lond 4 937

Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

Huygens C 1932 Oeuvres complmicroetes de Christiaan Huygens vol 17 pp 156189 The HagueMartinus Nijhoreg (Includes works from 16511666)

Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

Proc R Soc Lond A (2002)

578 M Bennett and others

regime exists only if the detuning is small enough consistent with our experimentalobservations Figure 6b shows the situation for shy xed cent as a function of iexcl and middot The anti-phase state sits in between the quasiperiodic and `deathrsquo states It is worthnoting that as we vary the platform weight M the path followed by our experi-ments is not strictly parallel to the middot -axis because varying M also mildly anotects thedimensionless damping iexcl (shy gure 6b)

Our results suggest that Huygensrsquos observation of `sympathyrsquo depended on bothtalent and luck The clock boxes were weighted by some 100 lb of lead in orderto keep them upright in stormy seas If they had not been the mass ratio wouldhave been too large making the coupling too strong and eventually stopping theclocks Neither would his observation have been possible if the coupling was tooweak since the small but inevitable dinoterence in the clock frequencies would preventfrequency locking Only clocks with sumacr ciently close frequencies could fall into anti-phase lock-step As it happened Huygensrsquos own inventions|and the clockmaker SOosterwijckrsquos craftsmanship|made such exquisite matching possible

We thank Don Aronson Chris Lobb Raj Roy and Steve Strogatz for their help over the courseof this work

References

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Burke J 1978 Connections Boston Little and Brown

Golubitsky M Stewart I Buono P-L amp Collins J J 1999 Symmetry in locomotor centralpattern generators and animal gaits Nature 401 693695

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Huygens C 1893 Oeuvres complmicroetes de Christiaan Huygens vol 5 pp 241262 The HagueMartinus Nijhoreg (Includes works from 1665)

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Huygens C 1986 Christiaan Huygensrsquos the pendulum clock or geometrical demonstrations con-cerning the motion of pendula as applied to clocks (translated by R Blackwell) Ames IAIowa State University Press

Korteweg D J 1906 Les horloges sympathiques de Huygens Archives Neerlandaises sparaer IItome XI pp 273295 The Hague Martinus Nijhoreg

Landes D S 1983 Revolution in time clocks and the making of the modern world CambridgeMA Belknap Harvard University Press

Lepschy A M Mian G A amp Viaro U 1992 Feedback control in ancient water and mechanicalclocks IEEE Trans Educ 35 310

Liao P amp York R A 1993 A new phase-shifterless beam-scanning technique using arrays ofcoupled oscillators IEEE Trans Microwave Theory Techniques 41 18101815

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Rawlings A L 1944 The science of clocks and watches New York Pitman

Rodriguez E George N Lachaux J-P Martinerie J Renault B amp Varela F J 1999Perceptionrsquo s shadow long-distance synchronization of human brain activity Nature 397430433

Strogatz S H amp Stewart I 1993 Scient Am 269 102

Yoder J G 1990 Unrolling time Christiaan Huygens and the mathematization of nature Cam-bridge University Press

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