An extension of Levi-Civita's theorem to nonholonomic dynamical systems

ACTA MECHANICA SINICA,Vol .6, No.2, May 1990 Science Press, Beijing,China Allerton Press,INC., New York,U.S.A.

I SSN 0567- 7718

AN EXTENSION OF LEVI-CIVITA' S THEOREM TO NONHOLONOMIC DYNAMICAL SYSTEMS

Naseer Ahmed

(Mathematics Department, Quaid-i- Azcan University, Ishmtabad, Pakistan )

ABSTRACT : This paper uses Poincar~ formalism to extend the Levi-Civita theorem to cope with nonholonomic systems admitting certain invariant relations whose equations of motion involve constraint multipliers. Sufficient conditions allowing such extension are obtained and,as an application of the theory a generalization of Routh's motion is presented.

KEY WORDS : nonholonomic dynamical systems, Levi-Civita's theorem, Poincar~ formalism.

I. INTRODUCTION

A theorem due to Levi-Civita Is'ill states that given a set of m invariant relations (m integrals ) in involution that are associated with an autonomous canonical system,one can determine a family of oom(oo 2re)particular solutions which confer a stationary value upon the Hamiltonian function of the system. Generalizations of this result have been studied in various directions in [ 1, 2, 3, 7, 9]. Re- cently, Capodanno I21 discussed it in generalised coordinates for nonholonomic systems admitting invariant relations which do not depend upon the time t.

With a view to exposing the scope of Levi-Civita's theorem, we shall study a variant of it tO cover the case of nonholonomic systems admitting m invariant relations whose equations of motion involve not only Suslove's constraint multipliers but also the Poincard parameters. We also establish the sufficient conditions which permit us to treat such systems.

Let r/p (oh,)be the independent Poincare parameters of real (possible)displacements of a dynamical system with n degrees of freedom whose position at any time t is specified by the variables xp(p = 1,2,..., n ); the group of operators

_ q c1 X p - r ~xq P ' q ' r = l ' 2 ' " " n (1.1)

define the infinitesimal displacements of the system in such a way that the variation dG(6G )of an arbitrary function G (xe, t ) in a real (possible)displacement of the system is determined by the formula

do-[-L-if- G +n X#ldt (fiG = ogvXvG ) (1.2)

Here d/ttt and Xp satisfy the commutative relations

(O/at ,Xp)=0 (Xp,Xq)=Cprq X r (1.3)

where C~q, being functions of the x 's, are skew-symmetric in the two indices p and q .

We assume that the system moves subject to differential (linear nonholonomic )constraints ex- pressed by I non-integrable equations

Received 29 July 1989

170 ACTA MECHANICA SINICA 1990

f~ (Xp, ~lp ,t ) = A,p%+ A~o=O (1.4)

where A ,p,A,o depend upon the x's and t, and the (o's satisfy the relations

Of, ogp=A,pogp=O (1.5) a~lp

Throughout this paper we follow the summation convention over the repeated indices with the following ranges: p, q, r, s = 1, 2,. . . , n ; a, p, p, v = 1, 2, ..., m < n ; a , 13, ~ = 1, 2,.-., l < n ;a, b, c = m+ 1,

As discussed in [ 4,6], let the system be completely characterized by the kinetic potential L (xp, qp, t )then its equations of motion in Poincare parameters are given by

d c3L dt aqp

aL _ X p L = )., af ~ (1.6)

where 2~ are the Lagrange undetermined multipliers. These equations can be put in the fo rm

d (a-'kp ~ ) cr aL -Xt, L-M, Xpf,

= M, (Ao~p+r/pA~p) (1.7)

where the quantities A~p, A~p, defined by the relations

aA~p , _ A~p- at XpA'~ Aqp-XqA,p'-XpA~q (1.8)

are skew-symmetric in the two-indices and Suslov's constraint multipliers M, are connected with 2~ by the relations /101

dM, = - 2 , d t (1.9)

Eqs. (1.7)are the Poincare equations involving the constraint multipliers.

II. CANONICAL EQUATIONS AND THE ASSOCIATED INTEGRALS

We introduce the function

H (Xp, yp, t)=yprlp-L

where the variables yp are defined by

(2.1)

aL af ~ yp= ~ +M~ ~ =gp(xq,~lq,t) (2.2)

We suppose that the Jacobian a (f, , gp )/a (M,, qp )#0, so that Eqs. ( 1 . 4 ) a n d (2 .2 )can be solved

to give

M~=M~(xp,yp , t ) ?/p= t/p (Xq, yq, t ) (2.3)

in view of which, the constraint Eqs, (I .4 )take the form

H, (xp , yp, t ) = 0 (2.4)

Vol.6 ,No. 2 Naseer Ahmed:An Extension of Levi-Civita's Theorem 171

Here H~ is the function obtained by replacing t/'s and M's by their values (2.3)in the functions f~ .

The Eqs. (l. 7 )can be transformed to the canonical form

t3H M~ ~H~ (2.5a) r / , = - ~ - y p - ~y,

J:e = -XpH+ M:XeH,+ c, rqrlqYr - M e r _ : _ �9 (Cqptlqh~r Aop qaA,q) (2.5b)

Assuming the constraints Eqs. (1.4) or (2.4)to be time independent, we have A~p=0. We set

K=H-M=H= a~q=CpqAar+A~,q (2.6)

and note that the constraint Eqs. (2.4) being non-integrable still hold along the trajectory so that

M= ~H= = c3 (M~,H~,) M=XpH~,=Xp(M~,H=) (2.7) gyp

Consequently, in view of the skew-symmetric property of the C~rq's and A~q's, the canonical Eqs. (2.5)

assume the form

dK qv = ~yp (2.8a)

yg = -Xp K Jc G~l~qYr'~- ap~t~q M, (2.8b)

where f~q ,being functions of x 's, are skew- symmetric in the indices p and q. These equations must be

supplemented by equations

dK xp= ~yq Xqxp (2.9)

which are obtained from (1.2)by taking G=xp and using (2.8a) . The equations of motion (2 .8 )o f

the nonholonomic system allow to determine the (n+ l)unknowns xl ,--. ,x,; M1 ,... ,Mi in conjunc-

tion with Eqs. (2.4)and (2.9). Throughout, the ' �9 ' will denote differentiation with respect to the

time t.

Concerning Eqs. (2.8)we remark that:

1. a sufficient condition for any function

, q~ (xp, yp) =constant (2.10)

to be an integral of the canonical Eqs. (2.8)is given by

t3r &P XpK+C~p-~p ~K ~, t3tp dr M,=O (2.11)

which is obtained by the use of (1.2)and is equivalent to

&p aK M =0 (2.12) (9,

172 ACTA MECHANICA SIN CA

Here (F, G )is the generalized Poisson bracket [61 defined by

1990

~3G X F dF dF dG ( F , G ) = ~ e - ~ X e G + C ~ e dy e ~q Yr (2.13)

The Eqs.(2.1 1 )or (2.12)is identically satisfied by taking r e , yp)= K (x e ,y e)and noting that

C~e = - C/q and ~ ~q = - f~qe ~ �9 Therefore, when H and H~ do not contain the time explicitly there

exists an integral

K = constant

2. the conditions of stationarity of K are

XeK=O ~K /dye=O (2.14)

These follow from the formula (1.2), giving

~K=coeXeK+ dK -~-~p'p ~Ye (2.15)

III. INVARIANT R E L A T I O N S

A relation of the type

F (xp, ye)=0 (3.1)

is defined [111 to be an invariant relation with respect to a given system of canonical equations if the

total time derivative of Fvanishes identically in virtue of the canonical equations of the system and

the relation itself. An invariant relation includes, as a particular case, an integral of the form

(2.10).

We observe that Eqs. (2.14)furnish 2n invariant relations that are associated with the canonical

Eqs. (2.8). For it can be easily verified that d (X e K)/dt and d (~K/~y e)/dt are linear combinations of

the quantities XeK and dK/dy e and vanish in virtue of Eqs. (2.8)and (2.14)themselves.

We now assume that the system of canonical Eqs. (2.8)together with (2.9)admits a set ofm dis-

tinct invariant relations of the form

F, (xp, yp)=0 (3.2)

which satisfy conditions analogous to (2.12), namely

dFr dFp M =0 (3.3) (Fo, ~'p)+f~pq gyp dyq

We now prove the following lemma.

Lemma In the presence of stationary nonholonomic constraints, if the system of canonical Eqs,

(2.8) admits m independent invariant relations (3.2)which can be solved for yg in the form

yu = ~p~ (xp , Ya ) (3.4)

Vol .6 , No. 2 Naseer Ahmed:An Extension of Levi-Civita's Theorem 173

then each of the functions cpu satisfies the identities

Xdpu_Xutp,+{tpu,tpv}+[LCuv+r C~ OtPVdya -r Oq)# ]Yr - C r a Z y a

_ M ~ ( ~ + O = 0r ~ OCu 0~ 0~pv)= 0 (3.5) #v - 'a t ~Ya --~"~av ~Ya - - ~ b Oy a ~Yb

where the braces { U, V }of the functions U(x a , Ya)' V(Xa' Ya)are defined by the relation I71 .

{ u , v } - ou ~v x~u+c~ ~u ~v --"--~a X a V - ~ OVa dyb Yr (3.6)

To prove the lemma, we first note that Eqs .(3.2 )yield an identity when from (3.4)y~ are replaced by their values r so that

XpFo + OF~ - (3.7) %-7 x,~,,=o

and OFa + OF~ c~ =0 (3.8) dy~ dyu dy~

Using the generalized Poisson brackets (2.13), the condition (3.3)can be rewritten as

OFp ~F~ OF# OFp +f2~q OF~ OFp dy e Xp F~ + r (3.9) X,F,+Cpq Ob Oyq dy9 dyq M~=O

Substituting from (3.7) the expressions for XeF ~ into (3.9) and breaking the sums over p and q from 1

to n into sums from 1 to m and (m + 1 ) to n, we find that

OF, OFprv ~ .. . 0~o, &ou &o~ Oy v ~y# L "e" v ~0#-- X# (Pv- Oy a Aa ~0# "r ~ X a (p#+ Crab ~Ya ~Yb Yr

+ ;v"{"C;a ay a -Cva-~ya)Yr-Met #v ~'~a# Oy a ~Ya ~Yb

which, in view of (3.6), reduces to

~y~ dy-T ~ - x. ~,~+ {q,., q~ } c L+ c L ~ - c;o ~ y~

( f U + ~ &Pv &P~ &Pu 0~, ) ] = 0 -M~ uv ~"~a~ OVa -~'~v Oy a --~ab ~a OYb

Since the functions Fl ,.-., Fn are independent, the non-vanishing of the Jacobian 0 (F I ,... ,F,n)/O (Yl, �9 .., y,, )leads to the required lemma.

IV. CONSTRUCTION OF FURTHER INVARIANT RELATIONS Since the system of canonical Equations (2.8) together with (2 .9) admits the invariant relations


(3.2), we have )~ ~ - - + ~ r OK +~et OK

_ dK dope' +C~a OK et Oq)~ OK M~ ~yq Xqq)l~- Tya XaK ~ Yr+~)as'~y a Oy,

We break the sums over q and s into sum~ over v and a and use (3 .6) to obtain

~KI-{K, ~la}'-F ~ ~,q" C,v'bCva'~y a Yr Met lay ~Jva"~y a

+ ~ c,:~y,-M, r~,:~-a~o-&Ty ~ (4.1)

This equation reduces to an identity when for each of the quantities yu we substitute the corre- sponding function (p~.

We replace y~ by their values ~pu in K and denote the resulting function by R* . Thus, from

it follows that

K (xp, y~ , ya)= R * (xp ' Ya ) (4.2)

0K X

XaK=XaR, - dK X ay-T a~

and OK _ dR* OK Oq~ ayo Oy. 0y# ay.

Relations (4 .3 - - 4 ), in view of (3.6) , yield

X~K+{K,q)~}=X~R" {R*, [ t + q,~}+ ~ x:,,+{~,~,~v

Substituting the expression for X~, K+{K, % } into (4.1)and using (4.5), we obtain

(4.3)

(4.4)

(4.5)

XIaR*"F{R*'~Is}-I- OR* [ r - ~ C# ayr Mal~'~et - ~ 0~# ) ] # a ~'~ba -~b

M~ ~ + et 0~pv dcPv _Da~ b =0 - ~'~#v ~'~a# dy a -~-~v dy a dy a dy b

In view of Lemma (3.5), the last result reduces to

_ _ r ' f , ,Oq,,)]OR" X#R*+ {R*,~pp }+lC#aYr-M~#a-~ba"-~y b =0 (4.6)

If the quantities f~ ~a vanish or r do not contain the independent Variables Ya ' the last result in either case may further be simplified to give

dR* X#R*4- {R*, tpp }+[C~aYr, M~O ~ a] ~ = 0 (4.7)

Vol.6, No. 2 Naseer Ahmed:An Extension of Levi-Civita's Theorem 175

We shall now show that the system of equations

Xa R * =- 0 63R-"'I* = 0 ~ya

are invariant with respect to canonical Equations (2.8) together with (2.9) First we notice that, in view of (4 .6)or (4.7), the relations (4.8)imply that

X~R*=0

Using Eqs. (2.8) it follows

(4.8)

(4.9)

d �9 . , 63 (XaR*)

63K XpXaR*.." 6 3 (XaR*)XbK+C~b ~K (SaR *)y r OYb

63(XaR*) 63K = M ~ ~ ~s 63y b 63ys

We split the sums over p, q, s into sums over #, b and c to obtain

* * * 0t d (XaR )={R ,X~R }+[]bcM~ 63(XaR*) 63R* dt 63yb 63Yc

+ ~ CAXrR'+X~176 ~ )

f r c~ ~ 63r ~ 63(XaR*)] + t c~byr-M~'ub+M'"cb-'-~c J ~Yb

Operating on each side of Ep (4.6)by X~,we find that

* * ( r 63q)~ )63(XaR*) XaX#R + {XaR ,tp~ }+ M~-~b+ M~fl ~ C~bYr- cb ~Yc 63Yb

=-{R*,Xatp~}---~b Xa [ , �9 + �9 63t9~,~ tCgbYr'M~['~#b M~['~cb"~yc )

Consequently

d-~(X, * * �9 , d 63R* R )={R ,XaR }+f~bcM,-'~yb (XaR*) t3y c

By a procedure similar to the preceding one can easily show that


Thus, each of d (X~ R *)/dt and d (dR */63y a ) / d t is a linear combination of XaR* and 3R*/63y a and therefore

Hence the relations (4.8)are invariant relations associated with the canonical Eqs. (2.8). From the preceding analysis we see that by means of a set ofm invariant relations (3.2)of the sys-

tem (2.8)which are solvable for y, in the form (3 .4 )and Satisfy lemma (3.5), we can construct further 2 ( n - m )invariant relations (4 .8)of the system of canonical Eqs. (2.8).

We remark that relations (4 .8 )and (4.9)confer a stationary value upon the Hamiltonian function. For the variation ~R*, computed in accordance with (1.2), is given by

�9 �9 �9 dR* t~R =(D#XlzR + (DaXaR + ~ a ~Ya

which vanishes as a consequence of the invariant relations (4.8)and (4.9).

V . G E N E R A L I Z E D L E V I ' C I V I T A ' S T H E O R E M

Let us now consider the nonholonomic system whose motion is determined by the canonical Eqs. (2.8) together with (2.9). We assume that the system (2.8) admits m invariant relations (3.2) which can be solved for y~ in the form (3.4)and satisfy lemma (3.5).

The Eqs. (2.8 b )and (2.9)consti tute a system of 2n first order ordinary differential equations which furnish the values of the variables Xp and yp as functions of the time t. We rewrite Eqs. (2 .8b) and (2.9) as

v . 1 + , ~ r t~K + ~ t~K �9 tgg Xqxa ;p=--e~.pl~ l,~qp"~yqy r L,!pq-~yqM~ (5.1a) Xa= tgyq

~ = t3K t~y""~ X q x tj (5.1b)

We now solve the new invariant relations (4.8) to obtain the values of 2 ( n - m ) variables x a and Ya in terms of the remaining m variables x~ .The substitution of these values into Eqs. (3.4)leads to determine the ( 2 n - m ) values of the variables x a and yp in terms of x, .We note that Eqs. (5. la ) are identically satisfied by these values. In view of the invariantive character of the relations (3.2)or (3.4) and (4.8), it follows that on substituting these values into Eqs. (5.1b), we obtain a system ofm independent equations, namely, which express ~, in terms of x~. The solution of this system which will contain m arbitrary constants will give oo m particular solutions of the canonical Eqs. (2.8)with (2.9).

I f the invariant relations (3 .2)are replaced by integrals of the canonical Eqs. (2 .8 )wi th (2.9) , they will contain another set of m arbitrary constants giving oo 2m particular solutions. Thus, we obtain the extension of Levi-Civita's theorem:

T h e o r e m If the system of canonical equations (2.8)with stationary nonholonomic constraints (1 A) admits a set of m independent invariant relations (m integrals )which are solvable with respect toy~ in the form (3.4) and satisfy the conditions (3.3) or (3.5), then the system has oom(oo 2m )particular solutions which are obtained by integrating a set of m first order ordinary differential equations.

If there is only one invariant relation (one integral)then by taking account of it we can construct further 2 ( n - 1 )invariant relations (integrals). This immediately leads to the following:

Corollary If the canonical equations (2 ~) with stationary nonholormmic constraint (1.4) admit one invariant relation (one integral )which can be solved for one of the yp's, then it has oo I (~2)particular solutions which are obtained by a single quadrature.

In the special case when there are no constraints, the x's are Lagrangian coordinates and r/p = ~p; Xp = c3/Oxp and all the Cqp'S vanish, our result subsumes the original theorem [s All

Vol. 6, No. 2 Naseer Ahmed:An Extension of Levi-Civita's Theorem 177

VI. GENERALIZATION OF ROUTH ~S M O T I O N Let us now consider the nonholonomic system with n degrees of freedom and the Hamiltonian

H (x v, yp),where 2n quantities xp and yp describe the state of the system.The motion of the system is determined by the canonical Eqs. (2.8)in conjunction with Eqs. (2.9).

We assume that the coefficients A~p and A~o occuring in the constraint Eqs. (1 .4) are such that A~o = 0 and A~n= A~p (X a ) . Consequently

XuH~=O (6.1)

Following the viewpoint of paper [ 5], let the displacement operators X~, X2 , . - . , Xm (m < n ) be cyclic, so that we have

(a) X.K=O

(b) (X. , Xq )=O

Moreover, in view of the relations (1.8), we have

Ot - - ~ _ _ (c) A~,~- A ~ - O

(d) a~,, = -A~u = - X a (,4 ~ )

and

(e) A]bare functions ofxm+ I , . . , , x n

The conditions (a) and (b) imply that C~q = 0 (q, r = 1,2, ... ,n ; a , b = m + 1, . . . , n ). Further we note that while Eqs. (2.8 a ) remain unchanged under conditions (a) - - (e), the Eqs. (2.8b) reduce to

yu=qaXag/u (6.2)

where ~bu = - M~A~u=~u (xb ) (6.3)

But in accordance with (1.2), we have

From (6.2) and (6.4) it follows that

whence

~u=rlaXa~u (x b ) (6.4)

L=q;.

Yu=~ku (x m+, , '" , Xn ) + c u (6.5)

where el, r ,'" ", Cm are arbitrary constants. We now introduce a function R, which, like the Routhian function, is obtained from K on

replacing y~ 's by their values (6.5). Then from equation

K (x a , ~1~+ c# 'Ya ) =R (x a , y , , cu )

We have OK = OR OYa ~Ya (6.6)

X a K = X a R _ OK Xa~k u (6.7) -67.

Consequently Equations (2.8)take the form

OK ya = - X a K + C r OK '~"- ,5'. q" Oy---~- Y"+n~q'tqM" (6.8)

178

and

and

ACTA MECHANICA SINICA 1990

#K ~a = ~ X~x a (6.9)

Breaking the sums over q and s into sums over # and b, Eqs. (6.8)and (6.9)assume the form

dK ~a- ~

dK d K ` +,,.,~ OK ,,. + ~ dK Ya= - X a g § "~b yr+Cr#a -~y# Yr ~l'ala dy~ - - 1VI ~ tlab ~ M~

�9 dK Xa= ~ y b XbXa

Using condition (b)and the Eqs. ( 6 � 9 9 ), we finally obtain

dR gla-- dYa

Ya = - X a R + C~a dR ~ dR Yr + aab-Sff

Which must be supplemented by the equations

dR Xa= ~ XbXa

(6.10 a )

(6.10b)

(6.11)

If the system of Eqs.(6.10b),in conjunction with (6.11 ),is integrable then we can find the values of 2 ( n - m ) variables x a , Ya" Substituting for x's into Eqs. (6.5), we obtain the values of (2n - m )variables x•+l,... ,x n and y! , . . . , Y n " Finally we can determine the values of the remaining variables xt, x2, ..., Xm by performing m quadratures of the equations

OK x~= ---~vXvx~

We have thus obtained a generalization of Routh's equations of motion td cover the case of nonholonomic system�9

We now consider the particular solutions of the nonholonomic system (2.8)together with (2.9) which possesses integrals (6�9 satisfy conditions ( a ) - - ( e ). Moreover the integrals (6.5)satisfy conditions (3.5). Accordingly, we can apply the Levi-Civita theorem to construct particular solutions. Precisely, the new invariant relations

XaR (xp,yp, % )= 0 dR (xp, yp, c~ ) dya = 0

furnish 2 ( n - m ) constant values of x a ,ya ;the Eqs.(6.5 ) then give Y~ as constants. Finally, we obtain ~

as constants and hence x, can be determined as functions of time t.

The particular solutions obtained here present a generalization of Routh's motion. In [ 1], an analogous treatment is given by Capodanno for gyroscopic systems in generalized coordinates.

REFERENCES [ 1 ] Capodanno, P., Sur denx extensions d 'un theorem de Levi-Civita aux syst&nes gyroscopiques et quelques applications, Jot~

hal d~m~canique, 17,3 (1978), 4 3 3 - 454. [ 2] Capodanno, P . , Extension of the Levi-Civita theorem to nonholonomic systems, Prikl. Math., Mech. 45, 3 (1981),

456 - - 465. or J. Appl. Math., Mech. 45, 3(1981 ), 332-- 338. [ 3 ] Dovletov, M. & Sulgin, M . F . , Particular solutions of dynamic equations with impulsive factors ofSuslov, G . K . , Taskent.

Gos. Univ. Nauen. Vyp., 397(1971 ), 60- - 66.

Vol. 6 , No. 2 Naseer Ahmed: An Extension of Levi-Civita's Theorem 179

[ 4] Ghori, Q. K. & Hussain, M., Generalization of the Hamilton-Jacobi Theorem, Z. Angew. Math. , Phys., 25(1974), 5 3 6 - 540.

[ 5 ] Ghori, Q . K . , Chaplygin's theorem of reducing multipliers in P oincar~- C hetaev variables, Z Angew. Math., Mech., 56, 4(1976), 147-- 152.

[ 6] Equations of motion in Poinca~-Cetaev variables with constraint multipliers, Bull. Austral. Math., Soc., 14, 4(1977), 3 2 7 - 337.

[ 7 ] Ghoti, Q. K. & Armed, N. , Levi~Civita's theorem for dynamical equations with constraint multipliers, Arch. Rational. Mech. Anal., 85, 1(1984), 1 - - 13.

[ 8 ] Levi-Civita I.,& Amaldi, U., LezJoni de meccanica razionale, 2, parte 2, Bologna, Zanichelli (1952). [9 ] Pignedoli, A., Ricerea di soluzioni particolari di un sistema anolonomo in base alia esistenza di integrali o relazioni

invarianti, Atti. Sem. Mat. Fis. Univ. Modena 1(1947), 95 - - 118. [ 10] Suslov ,G,K.[ F. K. Cycaos] ,Theoretical Mechanics [ TeopeTHqeci~aa Mexanmca] Fizmatgiz. Moscow (1946). [ 11] Whittaker, E.T, , A Treatise on Analytical Dynamics of Particles and Rigid Bodies, C. U. P. 4th edition ( 1952 ).

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An extension of Levi-Civita's theorem to nonholonomic dynamical systems