ACTA MECHANICA SINICA, Vol.9, No.2 May 1993 Science Press, Beijing, China Allerton Press, INC., New York, U.S.A.

ISSN 0567- 7718

C A N O N I C A L F O R M S O F N I E L S E N 'S A N D

C E N O V ' S D Y N A M I C A L E Q U A T I O N S "

Naseer Ahmed (Mathematics Department, Quaid-i-Azam University, lslamabad, Pakistan )

�9 /, ABSTRACT : In this paper, with Pomcare s formalism, and an indirect method, the canonical forms of the generalized equations of motion due to Nielsen and Cenov of a holonomic dynamical system in the

�9 t velocity-phase space and the acceleration-phase space are obtained in terms of the Pmncare parameters,

KEY W O R D S : holonomic dynamic system, Nielsen's equations of mot ion . Poincare"s equations of mot ion .

I. INTRODUCTION

In [11] and [ 2 - - 3 ] , using Lagrangian variables, Nielsen and Cenov have transformed equations of motion of a holonomie dynamical system to forms which involve the first order and second order time-derivatives of the kinetic energy of the system, respectively. Extensions of these results to nonholonomic systems containing higher order time-derivatives of the kinetic energy have been discussed in [ 4 - - 6] and [ 8 - - 10] , but no attention is paid to the transforma- tion of these equations to their canonical forms.

In a recent paper [~1, the author has studied the equivalent Hamiltonian forms of these equations. With a view to generalizing these results, we do not remain within the context of the usual Lagrange's theory of equations in generalized coordinates and examine the results given in [ 1] from the standpoint of Pomcare s theory of equations in group variables it31 . In fact, we have first derived Nielsen's and Cenov's equations of motion in terms of the first and second or- der time-derivatives of the kinetic potential function of the group variables, the Poincare parameters and possibly the time t . Final ly, with certain processes of variation, the dynamical equations so obtained are then transformed to their canonical forms in the velocity-phase space and in the acceleration-phase space.

II. PRELIMINARIES

We consider a holonomic dynamical system of N-particles whose positions at any time t are specified by rectangular cartesian coordinates ui ( i=1 , 2, . . . , 3N) . If m i = m i - l = m l - 2 denotes the mass of the i-th particle; ffi and U i , the components of the acceleration and force along the direction of us, then the fundamental equation of dynamics in view of the d'Alembert-Lagrange principle is given by [12] :

3N

= ( m i f f i - U i ) ~ u i = O (2.1)

where 6 u i 's represent the virtual displacements of the system. As discussed in [13] , we take x l , x 2 , " ' , xn as the independent variables to define the

configuration of the system such that

U i = l g i ( X l , X 2 , " " , X n , t ) = U i ( X p , t ) (/9=1, 2, . . . , n) (2.2)

then Eq. (2.1), with the summation convention, can be put in the form I71 :

Received 8 February 1991 * This paper was presented at the International Congress of Mathematicians ( I C M ) , 21 - - 29 August, 1990, Kyoto

University, Japan.

172 ACTA MECHANICA SINICA 1993

d ST _C8 ST C ' ST ] d[ O~p P O~------q " "qp~lq-~ r -XpT-Qp_cop=O ( p , q , r - I ,2,. . . ,n) (2.3)

Here the kinetic energy T is a function of the time, the xp s and the independent Pomcare parameters t/p' s ; cop' s are parameters of virtual displacement; the coefficients C~p and C'qp, be- ing functions of the xp's and the time t , are defined by the commutation relations

(X 0 , Xp)mqpXq (Yp, Xq)=C;qYr (2.4)

where the operators a

~ XP= ~q (Xr) ~Xq (2.5) X 0 = "~- -t" ~PO (Xq) OXp

fornfing a transitive group, define the infinitesimal displacements of the system. Moreover the variation dG (3G) of an arbitrary function G (xp, t ) in a real (virtual)

displacement of the system is given by the formulae:

dG=[XoG+qrXvG] dt (3G=~opXpG) (2.6)

Throughout our discussion, the repeated indices imply summation with the range of values: p , q,. r = 1 , 2, . . . , n , and dot is used to denote the differentiation with respect to the time t in accordance with (2.6).

IH. NIELSEN 'S AND CENOV'S EQUATIONS OF MOTION IN TERMS OF POINCARE'S PARAMETERS

Since the parameters o~'s in (2.3) are independent, it follows that

d 8T ST , ST -XpT =Qp (3.1) dt Orlp - cqp Otlq "- Cqp r I q S?lr

which are the Poincare:'s equations of motion for a holonomic dynamical system. We now trans- form Eqs .(3.1 ) to involve first and second order time-derivatives of the kinetic energy of the system. For this purpose, we need some important results which are given as follows:

LEMMA : For any function F (t, Xp, qp ) of class C2 in the domain of its arguments, the relations

SF /Oqp= Oi /Srip= Si/S~Ip (3.2)

d d (8f,/dqp) = SF/cgrlp- Xp F (3.3) d-7 (SF/OqP)= "-~

and

2 d (aF/8~p)-~ 8~'/8"r/p- X:,F (3.4) hold. dt

For the proof of the Lemma, see Ref.[8] . However, it is remarkable to note that O/Sqp and the operator Xp commute whereas U0qp and d/dt do not commute for any function of the xp's, gr's and the time t.

We take F = T in the relation (3,3) and substitute the result so obtained into Eq .(2.3) and note the independence of the parameters ~o?' s to obtain

-2~pT~-(C q Cqrp~q =Qp (3.5)

which give a generalization of the Nielsen equations of motion. Again, making use of the identity (3.2) in conjunction with (3 .4) , it can be immediately

shown thai

( ~ _3XpTX~ 2(g_, q ~71" ' } - ~ ,.. 0p --~q + C~prl q -~-~-~r ) = 2Qp (3.6)

Vol.9, No.2 N. Ahmed: Nielsen's and Cenov's Dynamical Equations 173

Eqs .(3.6) represent a generalization of the Cenov equations of motion. In the case when the system admits of the kinetic potential function L( t , x~, qp) defined

by the relation

L ( t , x p , tip)=T(t, xp, qp)-.V(xp) (3.7)

where V, being function of the xp's, is the potential energy of the system such that

Qp= -XpV (3.8)

then Eqs. (3.1) become

d t3L t~L r OL d t Otip C qP Otiq -- C q pti q (~l r--

I y p L ~ O (3.9)

Taking into account the Lemma ( 3 . 2 - - 4) with F=L, Eqs (3.,5) and (3 .6 ) take the forms :

_ q OL ~ = X e V (3.10) qP q ~?]r

and

C ~ OL )=XpV (3.11) ( ~ - 3 XrL ) - 2 ( C qr StiL"q + q P ti q "~ r

respectively. Thbse are the generalized forms of the Nielsen and Cenov equations of motion in t e r m s of the first and second order time-derivatives of the kinetic potential involving the

�9 l

Pomcare parameters. In the special case when the x/s are the Lagrangian variables and the qp's are the genera-

lized velocities Xp, the operators Xo, X e become O/Ot, O/Oxp, the coefficients c0qp, Cqp all vanish and E q s . ( 3 . 5 ) a n d ( 3 . 6 ) are reduced to equations obtained in [11] and [4] by Nielsen and Cenov, respectively, while Eqs .(3.10) and (3.11) become identical with those estab- lished in [1] .

IV. CANONICAL FORM OF EQUATIONS OF MOTION

Here we shall rewrite Eqs .(3.10) and (3,11) into their canonical forms. For this purpose, we have to consider the motion of the system in phase space together with a process of varia- tion of the set Of variables associated with the corresponding space. But from the forms of these equations it is evident that the sets of canonical variables are different for each set of. equation, which means that the processes of variation will also be different. In the sequel, we shall take up the cases of these equations simultaneously,

Let us first cohsider the motion of the system i~ the phase space. We take qp and the gene- ralized momenta yp , namely

yp = OL/~ip (4.1) as the canonical variables and introduce the following definition:

Definition 1 A set of 2n independent variables (xp, yp ) is said to form a phase space if

3t=0 6xp#0 6tip# 0 6yp~0 (4.2)

It is tO be remarked that 6xp#O, in view of (2.6), implies that ~op#0. Following the procedure a s discussed in [13] or [15], we introduce the Hamiltonian

function by the relation

H(t , xp, yp)=tipyp-L(t, xp, tie) (4.3)

and if, in view of (2.6), (3 .9 ) , ( 4 . 1 - 2 ) , we perform the &variation of (4.3) in accordance with Definition 1, Eqs .(3.9) can be transformed to the canonical form [13].

OH yp-- c~rl p f p= - XpH + Cqry+ Cq~ rlqY r (4.4)

with which we have to adjoin

174 ACTA MECHANICA SINICA 1993

Xp=?lqXqXp[- XoXp (4.5)

that are obtained by taking G=xp in (2.6). Turning to our main problem, we note that the comparison of Eqs .(3.10) or (3.11) with

Eqs.(3 .9) leads to the fact that Eqs . (3 .10) or (3 .11) , as they stand, cannot be brought directly to the form similar to that of Eqs. (4.4) by employing the &variation according to defi- nitions 2 or 3 as Eqs .(3.9) are transformed to Eqs .(4.4) by using definition 1. This happens due to the non-commutativity of d/dt and O/&lr for the function L (t, Xp, rip).

Thus, to overcome this difficulty, we follow an indirect method to bring Eqs . (3 .10) and (3.11) to their canonical form like (4 .4 ) . For this purpose, we take into account the formulas ( 3 . 3 - 4) with F = L and write Eqs .(3.10 ) and (3.11) into the following equivalent forms:

d aL aZ aL dI OIip up O?i'~-q -Cqpt]q~ -XpL=O (4.6)

and

d ~3i, OL _C r rl OL _XpL=O (4.7) all ~ "Cqp OO'q qP q Off----~

where the use is also made of the relations OV/Oqp=Ol,;/O~lp=OF'/Oqp=O and Xy=Ol)/&l= OI~/O~/which are obtained from (3.2) and (3.4) , respectively, by taking F=V(xp).

We are now in a position to attack the task of transforming the generalized Nielsen and Cenov equations of motion to the desired canonical forms.

As remarked earlier, we introduce the generalized momenta y ; and yp, corresponding to the set of Eqs .(3.10) and (3.11), by means of equations of transformation

and yp = 0/~/0q~ ( 4 . 8 )

fp = OL/O//'p (4.9)

Also, noting the facts that the xp 's are infinitely differentiable functions of the time t and each r/, is a linear function of the velocities jpt131, we introduce the following definitions:

Def in i t ion 2 A space of 2n independent quantities (qp, yp ) is defined as the velocity-phase space provided that

6t=O 6xp=0 6qp#O 6~p--AO 6yT--AO (4.10)

Definition 3 If the set of 2n independent quantities (Op, jTp) satisfies the conditions

tSt-~O 6xp=O &/~,= 0 6qp :# 0 6 ~ r 6y-p :~ 0 (4.11)

then we call it the acceleration-phase space. Let us consider the motion of the system in the velocity-phase space of the parameters r/, 's

and y p ' s and in the accelerat!on-phase space of the variables ~ p ' s and y~ ' s , respeotively. On the basis of Legendre s transformations, we adopt the standard procedure 1141 and intro-

duce the functions H*(t, YT' q p) and H ( t , 37p, j r ) analogous to the function ( 4 . 8 ) by the relations

H~(t,y~,, qp)=y~lp-s xp, "rip, qp) (4.12) and

/4( t , yp, ~p)=yp~p-L(t, Xp, qp, ~p, ~p) (4.13)

since the kinetic potential L depends upon the parameters xp ' s , ~/p's and possibly the time t . The functions given by (4.12) and (4.13) play the cole equivalent to the Hamiltonian function of the system in the velocity- and acceleration-phase spaces, respectively.

In conjunction with Definitions 2 and 3 we now perform &variations of (4.12) and (4.13) according to (4.10) and (4.11) to obtain

. �9 �9 �9 aZ aZ 6H =ypath,+rlpay 7- ~ &lp-

Vol.9, No.2 N. Ahmed: Nielsen's and Cenov's Dynamical Equations i75

and aL"

respectively. By virtue of the relations (4.8) and (4.9), the underscored terms in the last two expressions cancel each other and we are left with the following:

3//*= qp 6yp- (og/oqp)6q~, (4.14)

and 6 h = ~l'r 6 y p- (cgL /Oq p )6~l p (4.15)

which give the variations of the functions H * and H . Taking into account the equations of motion (4.6) and (4.7) together with the relations

(4.8) and (4.9) and using Lemmas (3.3) and (3.4), it can be immediately shown that

_ " * * r * OL/O~lp-2yp-C~pyr C~rqqYr

3y . Cq ,lqL

in view of which, the variational Eqs .(4.14) and (4.15) respectively, assume the forms

6H *= rip6y* p- (2y~- q * ' * " Copyq-Cqptlqy, )6rip

~t-I = il~OYp- (3yp- Coqp ; q - C qp rl q f , )6fl p

On the other hand, the variations 3H* and 6/t of the functions H ( t , ; p , qp) can be found as

OH *= (6H

From Eqs .(4.16 ) and (4.18) it follows that

*/Oy ;)Oy ~ + (6H */Orl r )6rip

/6;p )6y r + (3H /Otl, )6~lp

(4.16)

(4.17)

H * ( t , yp , tip) and

(4.18)

(4.19)

-C~,yq-C~Sqqy, 6q,=O \ \ o,7, Since the variations 6yp and 6tip of the independent parameters.y;'s and qp' s are arbitrary,

the last result yields

�9 OH** ., 1 @H * riP- -'~'~:- YP- 2 C3rlp +(CqPY~+CqPqqY*)/2 (4.20.)

which represent the canonical form of Nielsen's generalized equations of motion (3.10) in terms of Pomcare s parameters.

Again, in view of the independence of the variations 6qp and 6yp, (4.17) and (4.19) lead to the equations

Oy~ Y " = - 30~----~ +(C~Pyq+CqptlqYr)/3 (4.21)

�9 !

The set of Eqs .(4.21 ) is the required canonical form in terms of the Pomcare parameters of the generalized Cenov's equation of motion (3 .11 ) .

In the special case when the x, 's are the Lagrangian coordinates and the qp ' s are the generalized velocities ~, 's, the operators X o , Xp become c3/t3t, O/OXp; the coefficients c0qp, Cqp all vanish and Eqs . ( 4 . 2 0 ) and (4. 21) become identical with the results established in [ 1] .

Finally, we remark that in obtaining Eqs .(3.10) and (3.11) in forms (4.20Land (4.21), we have expressed Eqs.(3.10) and (3.11) into equivalent forms (4,6) and (4.7) and then de- rived them in the canonical forms. Thus , we have followed an indirect way and found

176 ACTA MECHANICA SINICA

E q s . ( 4 . 2 0 ) a n d ( 4 . 2 1 ) v ia E q s . ( 4 . 6 ) a n d ( 4 . 7 ) , r e s p e c t i v e l y .

1993

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