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D'Alembert's principle of zero virtual power in classical mechanics revisited

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1993 Eur. J. Phys. 14 217

(http://iopscience.iop.org/0143-0807/14/5/005)

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2/6

Eur.

J. P h p

14

(19931

211-221

Printed /n the UK

217

DAlemberts principle

of

zero

virtual power in classical

mechanics revisited

F

Verheestt and

N

Van den Bergh

tsterren kun dig Observatorium, Vakgroep Wiskundige Natu urk und e en Sterrenkunde, Universiteit Gent,

Krijgslaan 281, B-9000 Gent, Belgium

A+Dienst van de Studie-inspecteur, Fakulteit Toegepaste Wetenschappen, Universiteit Gent, Jozef

Plateaustraat

22,

B-9000 Gen t, Belgium

Received 24 May 1993

A b m a c r D l e m b e n s pnnciplc of anal)ucal mechmcs,

that the

vinual power of

he cansuaint forces

vanishes, c3n

be uwd to

determine

rclction, expleitl}

and umquel}. The

u l l equrralencc b e i w c n the Sewtoman, dAlcmbeman

and

Lsgrangiao

dcscnption;

is also

pro\&

1. Introduction

Most of the textbooks of classical mechanics, after

having exposed the Newtonian framework, use the

method of virtual work or power to introduce a

Lagrangian description of classical mechanics

with

constraints. In what follows we will give the logical

sequence, as we see it, and indicate some weaknesses

in the existing treatments.

Since the basic information

for

a given mechanical

problem

is

the knowledge abo ut the given forces and

the prescribed constraints, a Newtonian description

requires

the introduction

of

reaction

or

constraint

forces,

so

that the system obeys the constraints. That

constraints must give rise to forces is obvious,

because from a Newtonian point of view forces are

the only agents which can change the motion

of

the

system. However, the reaction forces are not given

in the same priori way as the external forces are,

but rather have to be treated as unknowns of the

problem. From this working definition the unique-

ness of the reactions is

not

evident.

Next comes the key notion of

virtual

velocities (or

displacements, to use the better known way of pre-

senting what ar e in reality aspects of calculus

of

vari-

ations). The observation, that in problems involving

equality constraints the total m echanical power asso-

ciated with the reactions usually vanishes for virtual

ResumS.

Le principe dAlembert en mkniquc classique,

a m m e

quoi la

puissance virtuelle

des forces de

contrainte

est nulle, pennet de determiner les &actions

de

fqon

explicite et unique. En outre, I uivalence totale

entre les

descriptions Newtonienne, dAlembertieone et

Lagrangienne est aussi prouvee.

velocities @ut no t fo r real motions in general), is then

erected into a principle. This principle sometimes

car

ries the name ofdAlembert (Rosenberg 1977, p 122),

although in many cases something different is meant

by dAlemberts principle. It is then claimed that the

said principle of zero virtual power determines the

reactions uniquely, with the help of Lag rangian m ul-

tipliers (pars 1965, p 24). although this assertion is

seldom fully proved, in the sense that ways ar e indi-

cated to determine the multipliers explicitly as func-

tions of positions, velocities and possibly time.

Furthermore,

th is

principle

of

zero virtual power

of

the reactions is mostly given, together with the

associated necessary fundamental equation, in the

case of statics. The transition to dynamics is then

done, more often than not, by the principle

of

educ-

ing dynamics to statics (Lanczos 1949, pp 88-92,

Goldstein 1980, pp 16-18). The latter principle is

also very often called dAlemberts principle. This

notion that one could somehow equate dynamics to

statics,

however,

is

confusing because

it

overlooks the

inherent difficulties in determining the

reactions

together with the motion of the system. In statics the

equilibrium positions are determined and the reaG

tions then merely balance the given forces, which is

an altogether different problem.

So

this part of the

usual textbook treatment is not satisfactory.


3/6

218

F Verheest

and

N

Van den

Bergh

I n addition, the fundamental equation (see (22)

below) following from the principle of zero virtual

reaction power (Pars 1965, p

28,

Rosenberg 1977,

p 149)

is

then used by many to introduce the Lagran-

gian description of motion with constraints, rather

than invoking a more abstract variational principle

like Hamilton's.

So,

before going

on

to

the Lagran-

gian description, one should prove that this funda-

mental equation is also sufficient to determine the

motion of the system, making afterwards the New to-

nian and Lagrangian descriptions totally equivalent.

This way back is also lacking in almost any treat-

ment involving virtual power.

In view of the sometimes critical remarks made in

the preceding paragraphs, we propose to revisit the

whole area around d'Alembert's principle, trying to

the only agents which can change the motion of the

particles (Rosenberg 1977, p 122).

The equations for the constraints themselves are

given in non-Pfaffian fom as

(3)

where A t and

E

are known functions of position

and time:

A t = A ; ( r l , , ,rN,f)

E m= B o ( r I , . , r ~ v ,) .

4)

Some of the constraints could be holonomic,of the

form

fa r l , .

..

r N , ) = o

5)

and their differentiated expressions are then under-

stood included in (3). Furthermore, the set (3) is SUP-

posed independent, i.e. reduced to the least possible

num ber (Pars 1965, pp 22-7). More precisely, the

rank of the matrix which one can construct from

(0= I , . . . ,L, 5

ive a systematically structured and complete

description of the transition from Newtonian to

~ ~ ~ ~ ~ ~ g i ~hanicr this route, including the

proofs for the uniqueness of the under

$Memb&'s p,.inhple an d the ull of

the different frameworks.

2. React ions and d'Alem bert 's pr in c iple

In

what follows we will only deal with equality con-

straints for systems of point masses, in order to keep

the notations and OUT l i e of approach sufficiently

transparent. When dealing with mechanical prob-

lems involving constrained motions, we are usually

given information about the constraints, as elabo-

rated further

on,

and about the external forces

Fh = F k ( r l , , . r N , l , . ., N,

)

k = 1,..., )

( 1 )

acting

on

point particles witb masses mk nd position

vectors rk with respect to an inertial reference system.

In

what follows,

k

will always be assumed to vary

from 1 to

N.

As a typical example to which we will

return later, the classical spherical pendulum is a

point mass moving under the influence of its weight

on the surface of

a

smooth, fixed sphere.

In order to fit the description of problems with

constraints into a Newtonian framework, originally

devised for one unconstrained particle, reaction

forces

Rk

are introduced, such that Newton's second

law reads as

mhrh Fh Rk.

(2)

The reactions

Rh

are not given in the same

a

priori

way as the external forces

Fk

are, however, but

rather have to be treated as unknowns of the pro-

blem. So in writing

Z),

one really defines the reac-

tions as those force terms which have to be added

to the equations of motion so as to make sure that

the components of

A ;

is

L.

Once the constraints are given, it becomes possible

to introduce at each point compatible with the holo-

nomic constraints

5 )

the set of virtual velocities w h,

which by definition obey

N

A t ,

wk

=

0 . (6)

Possible velocities on the other hand, being compat-

ible witb the constraints as they evolve in time, obey

3)

as already written down, Admissible trajectories

r h ( t )

satisfy 3 ) and

5).

The next step is to note that for all standard prob-

lems,

dealt with in the textbooks and involving equal-

ity constraints, the reactions globally do no work

under virtual displacements of the system. We will

call this d'Alembert's principle of virtual power,

and it states that at a given time t along trajectories

r h ( f ) he total m echanical power associated with the

reactions vanishes for any set

off

virtual velocities

( 'h).

k = l

(7)

Since the principle expressed in

7)

annot constitute

an additional constraint in view of the independence

of

(3),

the

Rk

have to be linear combinations of the

A t via Lagrangian multipliers A, giving

So

we see that the Newtonian equations of motion

can be rewritten as

the particles are obeying the constraints at all rel-

evant times. And the constraints must give rise to mhFh= F~ A , A ~ . (9)

forces, since in a Newtonian approach forces are

L

0 = 1


4/6

O AIembert s

principle

of zero

vir tual

power in ciassical mechanics

revisited 219

The com plete motion, i.e. the

N

unknowns rk(f)and

the newly introduced Lagrangian multipliers A e f ) , is

now to be determined from the N vector equations

(9) together with the L scalar equations 3,tarting

from initial conditions are adm itted by 3) and 5).

So far we have not stated any thing new o r exciting,

but have trodden common ground in recalling the

different notions involved (see e.g. Rosenberg 1977,

pp 119-48). We will now show what is usually

omitted, namely that it is possible to reduce the

above description further, by proving that the differ-

ent A,, and hence the reactions, can be determined

uniquely indeed as functions of positions, velocities

and time. T o

that

effect we arrange (9) along a given

trajectory as

which after scalar multiplication with Af and

s u m -

mation over

k

gives

(@= , ... L) . (11)

If we take the total time derivative of the constraints

(3), we find equations of the form

where the scalar expressions CQ re functions

C ( f )

C ' ( r ~ ( r )..., N ( t ) , L I ( r )...

N t ) ; f )

(13)

which we can explicitly compute. With the help of

(12) one can replace the first sum on the RHS of

(11). As the second s u m on the RHS of I

1)

similarly

is a function which follows from the given forces

and the constraints, (11) ultimately turns out to be

of the form

L

M @& = 09

o . = l

where

are the elements of a symmetric square matrix

M.

The expressions

D (f) =D (r l ( f ) ...,

N(f ) , r l ( f )

. . . ,

N f ) , f ) 16)

can be computed explicitly, as indicated above. N ow

the matrix M is positive definite, since

and the vanishing

of

the LHS

would

imply that

which in view of the independence of the set of con-

straint equations 3) is only possible for all U =

0.

Hence M is non-singular, so that (14) can be

inverted and the different A, uniquely determined,

although their expressions may be cumbersome

to

write down explicitly.

However, the reasoning in going from

( 6 )

to (18)

was done along trajectories, so that really we should

write

A )

=A&I ( .. . , r N ( f h ( 4 , .

. r N ( 4 , 4

(19)

for the fidlexpressions of the

A,

Ifw e now define new

functions

A.

through

= A a ( r l , .. . , N , l . . ,rN,

(20)

and write instead of (9) that

L

mkFk= F~ i ~;

or

we obtain N coupled ordinary differential equations

for the determination of the motion rk(t) of the

dynamical system. One can prove that the solutions

rk(f)of (21) are the same as of the set (3) and (9),

for given initial positions and velocities which are

admissible by the constraints. The preceding way of

presenting the equations

of

motion does not imply

that

i t

would be easier to work with (21) than with

(3) and (9) together, only that i t is possible concep-

tually to write the equations of motion as differential

equations for the determination of the trajectories

with given RHS.

Having dealt with the reactions, we deduce from

(2) and (7) or from (6) and (9) or from (21) a neces-

sary condition, sometimes called the fundamental

equation pars 1965, p 28, Rosenberg 1977, p 149),

valid for all admissible virtual velocities in points of

the trajectories considered. We have arrived at the

fundamental equation (22) without invoking the

principle whereby one is supposed to reduce

dynamics to statics, by calling the acceleration terms

the inertia forces (Lanczos 1949, pp 88-93). Th is

principle is often quoted as d'Alembert's principle,

although the notion that o ne could somehow equate

dynamics to statics is insulting to the genius of

d'illembert (Ham el 1949, p 220), since it totally

obfuscates the inherent difficulties in determining

the reactions together with the trajectories. In statics

the positions are known and the reactions then

merely balance the external or given forces. Noth ing

fancy there.

The fundamental equation (22) is

used

by many,

starting from a Newtonian point of view, to intro-

duce the Lagrangian description of motion with con-

straints, instead of invoking a more abstract

variational principle.

So

one also has to prove that

(22) is sufficient to determine the motion and hence

(18)


5/6

220 F

Verheest

and N Van

den

Bergh

the reactions, and it is precisely this way back which

is lacking in almost any treatment involving virtual

power or work, or deriving the Lagrangian from

the Newtonian formalism.

If

22)

is

not only neces-

sary but also sufficient to determine the motion,

then ultimately the Newtonian and Lagrangian

descriptions

will

be totally equivalent. This

is

briefly

addressed in the next section.

3. Sufficient conditions and equivalence

Here we start from a curve

rL(t) ,

obeying the con-

straints and such that

is valid for

all

virtual velocities (wh}. By

F;

and sub-

sequent an alogou s expressions with

a

prime we mean

tha t the relevant quantity

is

expressed along the curve

rL(t):

F; r) = F k ( r i ( f )

. . . ,

k( t ) , ? ' ( ( i ) . . . , h i),i). 24)

If we simply

call

then U) s in shorthand

S k = mti:

-

FL

U)

s,. Wk = 0

h = l

for all virtual velocities {wk} obeying

N

CAP.

h

=

0. (27)

k = 1

If at

this

stage one is happy to call Sk he reactions,

then the curve rL(t) corresponds to a motion of the

set of oarticles and the circle

is

triviallv closed

but also

A ? ( t i ) = ( t i )

B (rl)

=

P W (30)

(t i)

= f i h ) .

Since in the direct part we proved that the Lagran-

gian multipliers only depend on positions and veloc-

ities, there immediately follows that

&PI)

=

&(id

= P 11

(31)

Rk(fi)=Sk fd.

(32)

leading

to

the equality of

We can repeat the argument at other times and see

that

a

curve such that (23) holds for

all

admissible

virtual velocities indeed corresponds to

a

motion of

the dynamical system with given admissible initial

positions and velocities.

Once we have the

full

equivalence of the motions,

determined either from the reformulated Newtonian

equations of motion (21) or from the fundamental

equation

22),

the transformation

of

the latter into

Lagrange's equations of motion is standard and can

be found in the literature (see e.g. Pars 1965, pp

73-6, Rosenberg 1977, pp 201-6, Goldstein 1980.

pp 18-21, Gr iff iths 1985, pp 261-4).

4.

Spherical pendulum

As an illustration of the above remarks we will very

briefly recall the problem of the spherical pendu-

lum, treated

as a

point mass moving under the influ-

ence of gravity

on

the surface of

a smooth

fixed

sphere

wt

centre

in

and radius

e

Since the given

force is the constant weight of the particle, Newton's

second law reads as

m i =

mg

R. (33)

r . r = P

(34)

(Amold'1980, pp 91-2). However, we havd detiued

the reactions Rh

as

those forces which make the mov-

ing

system obey the constraints, and so there might

be subtle differences between the notions Rh along

a trajectory and Sk long a curve

so

far merely deter-

mined from (23).

The constraint is holonomic and SC~erOnOmic,

and in differentiated form

r . i . = O .

(35)

n

the latter &se, we repeat the argument exposed

in the previous section with Lagrangian multipliers

p m

and referring to the curve I;(?) to get

Along a trajectory r t ) the virtual velocities

w

have

to

obey

L

Sk = p e A P .

(28) r . w = O (36)

e.=I

indicating that they lie in the tangent plane to the

sphere, and d'Alembert's principle of virtual power

states that

Now we sta rt from ac urve with given initial positions

r l t J

and velocities

? ; t o ) ,

admissible in the sense of

R.w=O, (37)

g

compatible wit h &e constraints and obeying

(23)

at

all

times

i

to.

Then,

at

an arbitrary time

tl 4 ,

we use r;(t,) and

;;(ti) as

initid conditions

for traiectories

r J d .

solutions

of (21).

and see that

There being only on e vector constraint, we immedi-

ately find that the reaction


6/6

DAlemberts principle

of

zero virtual power

in

CIasSicaI mechanics revisited

22

equation of motion (33) is rewritten as

After scalar multiplication with r we can solve for X

and get

mi: =

mg+

Xr. (39)

(40)

r . r + i * = o (41)

(42)

m

r

X

=?(r.r-

r . g ) .

Taking the total time derivative of

(35)

gives

so

that 40) ecomes

A

= --(

2

i * + r . g ) .

Inserting this in (39) yields the differential equation

for the determination

of

the trajectory

r(f) .

One can

prove that the solutions of (43) are the same as of

the original set (35) and (39), for given admissible

initial positions and velocities. Moreover, the solu-

tions obey the constraint, as can be

seen

by tracing

the steps back from

(43) to (35).

5.

Conclusions

We have given and discussed in the preceding sec-

tions a logical sequence which leads from the Newto-

nian to the Lagrangian description of mechanical

problems with equality constraints, with the help of

the principle of dAlembert that the reaction forces

have no virtual power.

This

principle allows an

explicit and unique determination of the reactions,

with the full equivalence of the descriptions as a con-

sequence. We have indicated some of the weaknesses

in the existing treatments and try

to

remedy these.

In

all fairness it should be pointed out that the matter

under consideration here has recently been treated

using the methods of and couched in the language

of modem differential-geometric approaches to

mechanics (Cardin and Zinzotto

1989,

Giachetta

1992).

Acknowledgments

It is

a

pleasure to acknowledge the essential part

played by Professor

A

Vanderbauwhede almost a

decade ago in looking critically

at

the subject. We

would also like

to

thank Professors W Sarlet and

F

Cantrijn for their many incisive remarks which

undoubtedly improved our presentation beyond

recognition.

References

Arnold

V I 1980

Mathemtical Methodr

of Classical

Cardin

F and

Zanzotto G

1989 J .

Math. Phys.

30 1473-9

Giachetta G

1992

J. Math. Phys. 33

1652-65

Goldstein H

1980

Ch si co l Mechanics

2nd

edn (Reading,

Griffiths I B

1985

The Theory of/Clmsicol Dynamics

Hamel 949 Theorerisehe Mechanik Berlin: Springer)

Lanczos C

1949

The Variational Principles

of

Mechanics

Pars L A 1965 A Treatise on Anolyticnl Dynomics

Rosenberg R 1977

Analylicol Dynamics

of

Discrete

Mechanics 2nd edn (Berlin: Springer)

M A Addison-Wesley)

(Cambridge: Cambridge

University

Press)

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D'Alembert Principle of Zero Virtual Power in Classical Mechanics Revisited