New J. Phys. 18 (2016) 063018 doi:10.1088/1367-2630/18/6/063018

PAPER

Curl force dynamics: symmetries, chaos and constants of motion

MVBerry1,3 andPragya Shukla21 HHWills Physics Laboratory, Tyndall Avenue, Bristol BS8 1TL,UK2 Department of Physics, Indian Institute of Science, Kharagpur, India3 Author towhomany correspondence should be addressed.

E-mail: asymptotico@bristol.ac.uk and shukla@phy.iitkgp.ernet.in

Keywords:Newtonian, follower force, circulatory force, nonhamiltonian, vortices

Supplementarymaterial for this article is available online

AbstractThis is a theoretical study ofNewtonian trajectories governed by curl forces, i.e. position-dependentbut not derivable from a potential, investigating in particular the possible existence of conservedquantities. Although nonconservative and nonhamiltonian, curl forces are not dissipative becausevolume in the position–velocity state space is preserved. A physical example is the effective forcesexerted on small particles by light.When the force has rotational symmetry, for examplewhengenerated by an isolated optical vortex, particles spiral outwards and escape, evenwith an attractivegradient force, however strong.Without rotational symmetry, and for dynamics in the plane, the statespace is four-dimensional, and to search for possible constants ofmotionwe introduce theVolume ofsection: a numerical procedure, inwhich orbits are plotted as dots in a three-dimensional subspace.For some curl forces, e.g. optical fields with two opposite-strength vortices, the dots lie on a surface,indicating a hidden constant ofmotion. For other curl forces, e.g. those from four vortices, the dotsexplore clouds, in an unfamiliar kind of chaos, suggesting that no constant ofmotion exists. The curlforce dynamics generated by optical vortices could be studied experimentally.

1. Introduction

This concernsNewtonian dynamics driven by forcesF(r) depending on position r (but not velocity), whose curlis not zero so they are not derivable from a scalar potential [1]. Thus the acceleration (assuming unitmass forconvenience) is

( ) ( )= ´ ¹r F r F, 0. 1.1

Motion governed by curl forces is nonconservative: thework done byF(r) depends on the path. But, in contrastwith other nonconservative contexts such as velocity-dependent frictional forces, the force (1,1) is notdissipative. This is because [1] theflowpreserves volume in the position–velocity state space ( )�=r v r, : there areno attractors. In the absence of a potential, there is usually no underlying hamiltonian or lagrangian structure, soNoether’s theoremdoes not apply: the link between symmetries and conservation laws is broken, as elementaryexamples [1] demonstrate.

Most curl forces are nonhamiltonian (a special class [2], which is hamiltonian, will play no role in this paper).Without a hamiltonian, there is no conserved energy. Our aimhere is to investigate whether there are otherconserved functions of the variables (r, v). Alternatively stated, we ask about the dimensionality of regions instate space explored by typical orbits. The extreme case, ofmost interest, would bewhere there are no conservedquantities at all, andmotions explore regions of full dimensionality densely.Wewill not be able to answer thesequestions definitively using analytical arguments, but will present numerics indicating rich structures thatdeserve to be explored further. In particular, some curl forces F(r), apparently not special, indicate the existenceof (so far unidentified) constants ofmotion; and,more interesting otherF(r), also not apparently not special,indeed seem to explore regions of full dimensionality in the (r, v) state space, suggesting a type of chaos notpreviously encountered.

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13 June 2016

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Although the usefulness of curl forces as physicalmodels has been the subject of dispute in engineeringmathematics [3], their applicability in optics is not in doubt and their nonconservative nature has beenrecognized [4–9]. In this paper wewill use optical curl forces as examples, to illustrate themore general curl forcedynamics which is ourmain interest—even though it is known that not all curl forces can be realised optically(see appendix C of [2]).We consider the force on a small polarizable particle in amonochromatic lightfieldψ(r);if a is the ratio of imaginary and real parts of the polarizability [9], the optical force is proportional to

| | [ ] ( )*y y y= - +F a Im . 1.22

The second term is a curl force if

[ ] [ ] ( )* *y y y y´ = ´ ¹Im Im 0. 1.3

We should dispel a possible confusion. In optics, the term ‘curl force’ has sometimes [10, 11] been used in adifferent sense from (1.1), to denote forces that are the curl of a vector potentialA. To relate the twoterminologies, wefirst note that any F can be separated into its curl-free and divergence-free parts, givenrespectively by a scalar and a vector potential:

( )f= + = - = ´

´ = ⋅ =F F F F F A

F F

, where , ,

0, 0. 1.4grad curl grad curl

grad curl

In this representation, the curl condition (1.1) is

( )´ = ´ = ⋅ - ¹F F A A 0. 1.5curl2

When this holds, we call Fcurl a pure curl force.The separation (1.4) is not unique, because a gradient f1 can be added and subtracted from each part:

( )f f= - = +F F F F, . 1.6grad1 grad 1 curl1 curl 1

Tomaintain the separation,f1must be related to a new vector potentialA1 by

( )f f´ = = ´ ´ = ⋅ - =A A A A, i.e. 0, 0. 1.71 12

1 1 12

1

Formotion in the plane r=(x, y),A1 can be chosen to lie in the perpendicular direction ez, and itsmagnitude—the ‘stream function’A1—satisfies

( ) ( ) ( )f f f´ = ¶ -¶ = = ¶ ¶eA A A, , , 1.8z y x x y1 1 1 1 1 1

whose solution (see e.g. [12]) is

( ) ( ) ( ) ( )ò òf f= - ¢¶ ¢ + ¢¶ ¢A x y C x x y x y, d , 0 d , . 1.9x

y

y

x10

10

1

Anoptical curl force, defined by (1.2) and (1.3), is also a pure curl force, in the sense defined by (1.4) and(1.5), if the following condition holds (equivalent to the stationary continuity equation for theψ current):

[ ] [ ] ( ) ( )* *y y y y y y⋅ = = = A AIm Im 0, e.g. if real . 1.102 2

So, optical curl forces generated by fields satisfying theHelmholtz or Laplace equations are also pure curl forces.Our examples, illustrating whatwe think are general features of curl force dynamics, will be concernedwith

curl forces in the plane, usually generated by optical fields with one ormore vortices. The conservative (i.e.gradient-force) counterpart of this dynamics has been extensively studied (also in three dimensions) [13–17]; inaddition, aspects of the curl forcemotion have been studied, emphasising its azimuthal formnearvortices [18, 19].

We emphasise that although curl forces can be regarded asmathematically fundamental (themost generalcase of position-dependentNewtonian forces), they are not fundamental physically. They are effective forces—here, as elsewhere in physics, the inevitable result of idealisations. Our focus here is on the unexpected features ofthe dynamics generated by the simplest Newtonian force (1.1). So, rather than considering these idealisations indetail, we now comment briefly on several of them.

In (1.2), the curl force, giving rise to nonhamiltonian particlemotion, is associatedwith internal energydissipationwithin the particle (specified by the parameter a); part of this energy goes into radiation, but thisenergy, and its replenishment by the sources of the field, are not considered explicitly. Internal dissipation has acurious implication, discussed elsewhere [9]: in contrast with statisticalmechanics, wheremicroscopic elasticdynamics generatesmacroscopic friction, in the present casemicroscopic dissipation is associatedwithmacroscopicmotion that is not dissipative.

In a quantum treatment of optical forces, on atoms rather than classical polarizable particles [18, 19], thenonhamiltonian effective forces governing the centre-of-massmotion emerge from amore fundamentalhamiltonian formulation of the total system, based on the optical Bloch equations.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

Another idealisation, that we discussed in detail [20] andwhose violation involves additional forces, is theadiabatic separation between the slow translation and the fast internal dynamics of the particle. Furtheradditional forces arise for particles whose size, relative to the optical wavelength, is not small [21, 22].

The structure of the paper is as follows. Section 2 concerns forces with rotational symmetry.We show theunexpected result that in a large class of cases the existence of a curl forcemeans that the particle always escapesto infinity; even an arbitrarily strong accompanying attractive gradient force fails to restrain it.

Section 3 concerns forces that do not possess rotational symmetry. The state space (r, v) is four-dimensional.As a numerical tool for displaying possible constants ofmotion, we introduce (section 3.1) theVolume of section(VoS), by analogywith the Poincaré surface of section familiar in hamiltonian dynamics. TheVoS is a numericalprocedure, inwhich the state variables at times along an orbit which satisfy a particular condition (e.g. crossingthe x axis), are plotted as dots in the three-dimensional space of the remaining state variables.We give twoexamples of curl force dynamics associatedwith several optical vortices of zero total strength. In one (section 3.2)the dots lie on a surface, indicating the existence of a hidden constant ofmotion (hidden, in the sense that we donot know its functional form). In the other (section 3.3), the dots form a cloud, indicating no constants at all, i.e.orbits exploring a 4D region of the state space. This seems to be an unfamiliar kind of chaos. Readers interestedonly in this latter case should look at the ‘dust clouds’ infigure 9.

Questions raised by this study, and the possibility of experimentally detecting the new features we haveidentified, are discussed in the concluding section 4.

2. Rotational symmetry: separability, and inevitable escape froma single optical vortex

It is convenient towrite themost general rotationally symmetric dynamics in the plane r=(x, y)=r(cos θ,sin θ), of the type (1.1), in the form

( ) ( ) ( )= - + qr e eg rh r

r. 2.1r

Thefirst term is a radial gradient force (with potential given by the integral of g(r)), attractive if g(r)>0, and thesecond is an azimuthal curl force. In terms of the angularmomentum

( )�q=J r , 2.22

the dynamical equations for the evolution r(t) and J(t) are

( ) ( ) ( )�= - =rJ

rg r J h r, . 2.3

2

3

This is already a reduction from four freedoms in (1.1) (i.e. )x y v v, , ,x y to three (i.e. )�r J r, , , reflecting the factthat changing the starting azimuth θ(0) simply rotates the trajectory.We assume that h(r) has the same sign forall r—positive, say. Then (2.3) shows that the angularmomentum always increases—an obvious consequence ofthe torque associatedwith the azimuthal force, as well as illustrating the non-applicability ofNoether’s theorem(angularmomentumnot conserved, although there is rotational symmetry).

A further reduction is possible, to two freedoms. Generalizing our earlier analysis [1], to include the radialforce in (2.1), we transform the independent time variable t to J, sowe nowwrite r( J) instead of r(t).We have,using (2.3)

( )( )�

��

º ¢ = =r

Jr

r

J

r

h r

d

d. 2.4

Thuswe get the r( J) dynamics

( ) ( [ ( )]) ( ( ))( )

( )+ ¢¶¶

=-

r rr

h rJ r g r

h rlog . 2.52

2 3

2

Thus rotational symmetry has separated the dynamics, with reduction to two freedoms: ¢r r, : the θcoordinate has been eliminated.We canmake the r equation simpler by defining the new radial variable

( ) ( )òºR r h rd . 2.6r

r

1 10

Thuswe seek the dynamicsR( J) rather than r( J) or the original r(t). The relevant derivatives are

( ) ( ) ( ) ( ) ( )¢ = ¢ = + ¢¶¶

R r h r R r h r rr

h r, and , 2.72

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

leading to thefinal equation of radialmotion

( ( ))( )

( )=-

RJ r g r

h r. 2.8

2 3

Explicitly

( ) ( ( )) ( ) ( )= = =R R J r r R J RR J

J, ,

d

d. 2.9

2

2

Once (2.8) has been solved forR( J), r( J) can be determined by inverting (2.6). Then the time variable can bereinstated from the second equation in (2.3):

( )( ( ))

( )ò=t JJ

h r J

d. 2.10

J

J1

10

Thuswe have r(t) and J(t). Finally, the azimuth can be found from (2.2):

( ) ( )( )

( )òq =tt J t

r t

d. 2.11

t

t

0

1 12

1

This completes the separation of the radial and angular dynamics, reducing the original four-freedomsystem to two, i.e. ¢R R, .The reduced dynamical equation (2.8) is of hamiltonian form:

( ) ( ) ( )

( )( )

( ) ( )( )

ò

= = +

= + ¢ ¢

H R P J E J P U R J

U R JJ

r Rr g r

, ,1

2, , where

,2

d . 2.12r R

2

2

2 const.

Wenote that this is J dependent, and since J corresponds to time, the energyE is not conserved; in fact it alwaysincreases:

( ) ( ).=¶¶

=E J

J

U

J

J

r

d

d0. 2.13

The repulsive part J2/2r(R)2 of the potentialU(R) in (2.12) always increases. Although this argument isformulated in the (R, J) plane, in the original (r, t) plane it carries the unexpected consequence that the particlealways recedes from the origin, however strongly attractive the radial force g(r) is (exceptwhen it is a hardwall, inwhich case the particle spirals ever closer to thewall, ever faster).

An important special case of rotational dynamics is generated by optical forces from a single isotropic vortexof orderm, whose light wave is

( ) ( ) ( ) ( )y q= + =r x y r mi exp i . 2.14m m

According to (1.2), the radial and azimuthal forces in (2.1) are

( ) ( ) ( )= =-g r r h r ar, , 2.15m m2 1 2

and the curl force is a pure curl force according to (1.10). Thus (2.8) simplifies to

[ ]( )

( ) ( ) ( )r r r

r

= -

=+

=+

- - + - +B Jm

aR B

m

a

,

where2 1

,2 1

. 2.16

m m m2 2 3 2 1 1 2 1

2

/ /

This is a variant of the Emden-Fowler equation [23, 24], with two source terms instead of one; a general solutionin closed form seems unavailable. The variables are related by

( )( )�r r= =+ +r J a, , 2.17m m m2 1 2 2 1/

and the hamiltonian (2.12) simplifies to

( ) ( ) ( )

( ) ( ) ( )( )( )

r r

r rr

= = +

= + +

r r

- ++⎛

⎝⎜⎞⎠⎟

H p J E J p U J

U J B m Jm

, ,1

2, , where

,1

22 1 . 2.18m

m m

2

2 2 2 12 2 1

//

The potential, whose perpetual increase causes the particle to escape, is illustrated infigure 1.In an alternative expression of the dynamics (2.1), position is denoted by the complex variable

( )= +z x yi . 2.19

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

Now the evolution (1.1) is

( ( | |) ( | |) | | )| |

( )=- +

zg z h z z

zz

i. 2.20

In this representation, the one-vortex dynamics takes the formof a stationary nonlinear Schrödinger equation(with t analogous to a coordinate):

| | ( ) ( )= - +-z z a z1 i . 2.21m2 2

Although a general solution seems unavailable, a particular solution can be found, representing a particleescaping to infinity at time t=t0 while spiralling logarithmically (figure 2). Form>1, this is

( ) ( ( ( )))( )

( )( )=-

- -z t

C q t t t

t t

exp i log, 2.22

m0 0

01 1

inwhich

( )( )

( )( )

( )( )

=+ + + +

-=

+-

-⎛⎝⎜

⎞⎠⎟q

m m ma

a mC

q m

a m

1 1 4

2 1,

1

1. 2.24

m2 2 1 2 2

This solution corresponds to the initial conditions

( )( )

( ) ( ) ( ) ( ( ) )( )( )

( )( ) ( )= = + =+ -

-- - +x

C

ty v v

C m q

m t0 , 0 0, 0 i 0

1 i 1

1. 2.25

m x y m m q0

1 10

1 i

For the excluded casem=1, (2.21) is a linear equation, whose spiralling solutions are

( ) ( ) ( ) ( )= - + - -+ -z t c t a c t aexp i 1 i exp i 1 i . 2.26

Figure 1.Potential in (2.18) form=2 and different J.

Figure 2.One-vortex track (2.22), for (0�t�0.99t0), t0=1,m=2, a=1, corresponding to initial conditions (x0, y0, vx0,vy0)=(3.269, 0, 3.269, 11.642).

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

3.More optical vortices: seeking chaos

3.1. TheVoSWhat distinguishes dynamics under curl forces fromhamiltonian dynamics? A fundamental difference would bemotion exploring a full-dimension region of the (r, v) state space. Formotion in the r=(x, y) plane, this wouldbe exploration of a four-dimensional region. As a numerical tool for investigating this possibility, we introducetheVolume of sectionVoS, defined as follows.We select times tn along an orbit r(t) satisfying some condition, forexample x(tn)=0. At such times (identified numerically), we plot the other three variables, for example (y(tn),vx(tn), vy(tn)), and examine the dot patterns in this three-dimensional space—theVoS—after long times. Thereare four natural choices for theVoS, corresponding to x(tn)=0, y(tn)=0, vx(tn)=0, vy(tn)=0; wewill denotethese byVoSx, VoSy, VoSvx, VoSvy. Although the full dynamics in 4D state space is volume-preserving, theabsence of underlying symplectic structuremeans that the 3Dmap between successive dots on eachVoS doesnot preserve volume. This contrasts with the area-preserving 2DPoincarémap on a 3D constant-energyhypersurface in hamiltonian dynamics.

If there is one conserved quantity, as in a hamiltonian system, the dots in eachVoSwill lie on a surface, and ifthe dynamics is chaotic the surface will be partly orwhollyfilled. If there is an additional constant, as in anintegrable system, the dots will lie on a curve. And in the situationwe are contemplating, inwhich there is noconserved quantity, the dots willfill a volume.

3.2. Two vorticesTo explore the possibilities, we need to choose suitable forces F(r). Aswe have seen, the optical curl force from asingle vortex always leads to escape, even in the presence of an attractive gradient force. The escape is associatedwith the continuous increase of angularmomentum caused by the torque from the curl force. It is natural to tryto avoid this by exploring the curl force from two vortices of opposite strength, so at large distances the nettorque tends to zero. The simplest such opticalfield, representing two vortices on the x axis, at x=+1withstrength+1 and at x=−1with strength−1, is

( ) ( )( ) ( )y = - + + -r x y x y1 i 1 i . 3.11

This generates a curl force. Although it is not a pure curl force, because (see (1.10)) y = 4,21 it can easily be

made so, for example by adding−2y2.We note in passing that the associated gradient force

( ) | ( ) | { ( ) ( )} ( )y= - = - - +F r r x r y r4 1 , 1 , 3.21grad 12 2 2

is integrable aswell as hamiltonian. Of course the energy

( ) | ( ) | ( ) ( ) ( )y= + + = + + + -rE v v v v r x1

2

1

21 4 3.3x x x x1

2 21

2 2 2 2 2 2

is conserved. And, as can easily be confirmed, the following quantity is also conserved:

( ) ( ) ( )= ´ - - +r v eK v y r. 4 8 2 . 3.4z y12 2 2 2

Figure 3 shows the track of an orbit in the (x, y) plane, with a pattern clearly illustrating the integrability.These conserved quantities are destroyedwhen the gradient force is combinedwith the corresponding curl

force according to (1.2):

( ) { ( ) ( )} { } ( )= - - + + - - -+F r a x r y r a xy x y, 4 1 , 1 2 2 , 1 . 3.51 curl grad2 2 2 2

Figure 4 shows the pattern of directions of this force, asymptotically attractive and inspiralling to the two vortices(circles), with a stagnation point (square) on the negative y axis.

This is a promising choice for a curl force, because it fails the ‘anisotropic hamiltonian’ test derived elsewhere(equation (2.6)) in [2]. Figure 5 shows an orbit generated by this force, in the (x, y) and (vx, vy) (hodograph)planes. It appears irregular, indicating that the system is not integrable: there is atmost one constant ofmotion.Of course this cannot be energy because the force is non-conservative.

To investigate whether in fact there is such a constant ofmotion, we show infigure 6 the correspondingVoSpatterns. The dots clearly lie on surfaces, indicating that a constant ofmotion, that is, a conserved function ofrand v, associatedwith the force (3.5), does exist.We do not knowwhat this constant is: the complicated formofthe surfaces (including the holes, which do notfill up in simulations for longer times) suggests that it is not asimple function. Its existence raises the possibility that with a suitable change of variables the constant couldrepresent a hamiltonianH that generates themotion, as in (2.12) for the rotationally symmetric case—though inthis two-vortex case any suchHwould be time-independent.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

3.3. Four vorticesWehave investigated two-vortex fieldsmore general than (3.5), in which the gradient force is not of optical type,suggesting curl force dynamics with no constants ofmotion. But forces inwhich both the curl and gradient partsare optical are not only conceptually simple but could also be explored experimentally. For this reason, we nowconsider the optical force from four alternating-sign vortices arranged on a square:

( ( ))( ( ))( ( ))( ( )) ( )y = + + + + - - - + - - - +x y x y x y x y1 i 1 1 i 1 1 i 1 1 i 1 . 3.62

Againwe create the corresponding curl+gradient force according to (1.2):

( ( ) ( ))( ( ( ) ) ( ( ) )) ( )

=- + - + -

+ - + - - ++F x r x y x r y x

a x r x y y r y x

8 4 12 , 4 12

8 3 4 , 3 4 . 3.7

2 curl grad6 2 2 6 2 2

2 2 2 2 2 2

Figure 7 shows the pattern of directions of this force, inspiralling to the four vortices (circles), with threestagnation points (squares) on the x axis, and asymptotically attractive.

Figure 3.Orbit in (x, y) plane for the gradient (i.e. not curl) force F1grad (equation (3.2)) from two vortices, for 0�t�100 andwith(x0, y0, vx0, vy0)=(1/2, 1/2, 1/2, 1/2).

Figure 4.Two-vortex force F1curl+grad (equation (3.5)) for a=2.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

Figure 8 shows an orbit generated by this force, in the (x, y) and in the (vx, vy) (hodograph) planes. Aswith thetwo-vortex orbit infigure 5, this looks irregular, indicating that the dynamics is not integrable, and again raisingthe question of whether any constant ofmotion exists.

Figure 9 shows the correspondingVoS dot patterns for this four-vortex dynamics. They are very differentfrom the two-vortexVoS patterns infigure 6. It is hard to interpret these ‘dust cloud’ patterns as anything otherthan irregularly exploring a four-dimensional region in the state space. If correct, thismeans that curl forces cangenerate an unfamiliar kind of chaos—contrastingwith hamiltonian chaos, where because of energyconservation a hypersurface is explored. Preliminary computations support the conjecture that orbits withnearby initial conditions separate exponentially. The dust cloud patterns appear to possess a complicatedstructure; there are ‘holes’, almost or totally devoid of dots, which do notfill up in simulations for longer times;and some regions appear denser than others, almost hinting that the orbit would condense onto a chaoticattractor—which of course it cannot do because the full dynamics is 4D volume-preserving.

Irregular trajectories generated by curl forces, have been demonstrated in simulations before [16, 19] thoughnot emphasised as nonhamiltonian and for times too short to identify constants ofmotion.

4. Concluding remarks

This study reveals a variety of structures in orbits governed by rather simple curl forces, including those exertedon small particles fromoptical waveswith vortices. For a single optical vortex, the orbits spiral outwards andalways escape.When there are two vortices, numerics indicates a hidden constant ofmotion, even though thereis no conserved energy.Most interesting are other cases, for example four vortices, where it seems there are noconserved quantities at all. Further computations, not reported here, show that these different behaviours arenot exceptional.

We regard this as an exploratory study, raising several questions:

• Are the dust cloud patterns such as those infigure 9 typical in situationswhere orbits are bounded?

• How can their structures be characterised?

• For opticalfields with an infinite periodic array of vortices, with total strength zero in each unit cell, canmotion under the associated curl plus gradient forces be chaotic and explore the full state spacedimensionality—and, if so, is this typical or exceptional? (This would extend previous studies [14, 15] of orbitsin optical lattices under conservative forces.)

Figure 5.Orbit in the (x, y) plane (left) and the (vx, vy) (hodograph) plane (right) for the two-vortex force F1curl+gtad (equation (3.5)),for a=2, (x0, y0, vx0, vy0)=(1/2, 0,−1, 2), and 0�t�1000.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

Figure 6.Two-vortex volume of surface patterns corresponding to figure 5, for (0�t�50 000): (a)VoSx; (b)VoSy; (c)VoSvx; (d)VoSvy. To enable the surface to be visualisedmore clearly, we include separately in the supplementarymaterial figure 6e.wrl, bywhichthe VoSx pattern can bemanually rotated andmagnified.

Figure 7. Four-vortex forceF2curl+grad (equation (3.7)) for a=1.8.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

Figure 8.Orbit in the (x, y) plane (left) and the (vx, vy) (hodograph) plane (right) for the four-vortex force F2curl+gtad (equation (3.7)),for a=1.8, (x0, y0, vx0, vy0)=(0,0,−0.1,−0.1), and (0�t�75).

Figure 9. Four-vortex volume of surface ‘dust cloud’ patterns corresponding tofigure 8, for (0�t�10 000): (a)VoSx; (b)VoSy; (c)VoSvx; (d)VoSvy. To enable the dust cloud to be visualisedmore clearly, we also include separately in the supplementarymaterialfigure 9e.wrl, bywhich theVoSxpattern can bemanually rotated andmagnified.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla

• Are periodic orbits dense, as in hamiltonian systems? This is not a trivial question because, as we discussedelsewhere (section 4 of [2]), the nonconservative nature of curl forces imposes strong restrictions on the formsof periodic orbits.

• Where numerics strongly suggests a constant ofmotion, as in the case considered in section 3.2 and illustratedinfigure 6, is there any analytical way, general for curl forces (1.1), to establish its existence and characterise it?

• Can the curl force dynamics we have identified theoretically be seen experimentally, in themotion of smallpolarizable particles, governed by forces fromopticalfields with several vortices? This is not straightforward:to see theNewtonian particlemotion generated by (1.1), the particles would need to be trapped in a vacuum,unlike the viscosity-dominatedmotion inmany experiments [25–28]—which also emphasise effects of theangularmomentum, for example particle rotation, rather than the centre ofmassmotion considered here.

Acknowledgments

We thank three referees for their detailed reading of the paper, and their very helpful criticisms and suggestions.MVB’s research is supported by the LeverhulmeTrust.

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New J. Phys. 18 (2016) 063018 MVBerry and P Shukla