Journal of Applied Mathematics and Physics (ZAMP) Vol. 27, 1976 Birkh~iuser Verlag Basel

Hamilton-Jacobi Theorem in Group Variables

By Munawar Hussain, Mathematics Dept., Governmen t College, Lahore, Pakis tan

1. Introduction

Consider a conservative holonomic system with n degrees of freedom and whose position is defined by n variables x l , . . . , xn. Let T and U be the kinetic and potential energies of the system respectively.

We now use the Poincar6-Cetaev method [1, 6] to obtain the equations of motion of the system.

Let -ql, ~72 . . . . . % be the parameters of real displacement and X~, X z . . . . . X~ the corresponding displacement operators which are expressed by the relations

X , = ax j ( i , j = 1, 2, n), (1) t

where ~ are known functions of x~, x2 . . . . . xn and summation over a repeated suffix is understood. Since these operators form a group we have the commutation relations

(Xt, Xj) = C, j zXk , (i, j, k = 1, 2 . . . . . n). (2)

Here C, jz are constants of structure of the group and depend upon the choice of ~'s. As in [6] an infinitesimal real displacement of the system is defined by

] d f ( x l , x2 . . . . . xn; t ) = -~ + ~ ,X~( f ) dt . (3)

If L = T - U is the Lagrangian of the system, then the equations of motion of the system are �9

- Cm~?j ~ - X,(L) = 0, (i, j, k = 1, 2 . . . . . n).

By introducing the moment y~ by the relations

aL y, = - - , (i = 1, 2 . . . . . n), a,1,

the canonical equations of the system and Hamilton's partial differential equation as obtained in [1] are

dy, OH --~ = Cj,~Tjyk - X~(H) , ~7, = ay, (i, ], k 1, 2 . . . . . n), (4)

aV - - + H(x~ , x ~ , . . . , x , , X ~ ( V ) . . . . . X , ( V ) , t ) = O. (5) at

286 Munawar Hussain ZAMP

2. T h e H a m i l t o n - J a e o b i T h e o r e m

We shaU prove the following:

T h e o r e m . If V(xx, x2 . . . . . xn, ax, a 2 , . . . , an; t) is a complete integral o f Hamilton's partial differential equation (5), then the integrals o f Hamil ton's equations (4) are given by the equations

8V y, = X,(V), b, = - ~a,' (i = 1, 2 . . . . . n), (6)

where the b's are n new arbitrary constants.

Proof. We shall prove that the functions

x~ = x,(al, a2 . . . . . an; bl . . . . . b , ; t)'~ Yl y~(al, a2 . . . . . a ,; bx, ., b , ; t ) J (7)

determined from (6), satisfy the equations (4) for arbitrary values of a 's and b's. Now V satisfies (5) for all values of x's, a 's and t in the appropriate domain. So

substituting the complete integral in (5) and differentiating partially with respect to a,, we have

02 V OH 0 Oa,Ot + ~ ~a, (Xj(V)} = 0. (8)

Also the equation b~ = - ( 0 V/Oa~) is satisfied identically if we substitute for each x from (7). Substituting these values and differentiating with respect to the time t we get

0 Ot~at

Now, since V ~ C2 we have

O2V 02V OtOa~ Oa~Ot

o = ~ , (x~(v)).

Therefore (8) and (9) yield

.qj co//\ / 8 V \ = o .

(9)

Moreover, there are n such equations, one corresponding to each a~ and the determinant of the coefficients [Xj(OV/OaO[ is nonvanishing. Therefore

0H ~h = -0--~y ' ( i = 1, 2 . . . . . n). (10)

Now we again substitute the complete integral in (5) and apply the operator X~ to get

( o e ) OH X~X,(V) = 0. (11) x, -~ + X,(H) + oyj

The equation

y, = X,(rO

Vol. 27, 1976 Hamilton-Jacobi Theorem in Group Variables 287

is satisfied identically if we substitute for x 's and y ' s their values from (7). Substituting these values and differentiating with respect to the time t, we have

dy~ a d--7 = ~ {X,(V)} + ,TjXjX,(V). (12)

Now the relations (11) and (12), in view of (10), yield

dy, -~ = ~ j ( X j , X , ) V - X , ( H ) .

With the help of (2 )and (6) the last equation becomes

dy, --~ = ~tC~,kYk -- X~(H), ( i , j , k = 1, 2 . . . . . n). (13)

The relations (10) and (13) show that the x 's and y 's given by (7) satisfy Hamil ton 's equa- tions, and thus the theorem is proved.

References

[1] N.G.(~nTAI/v, On the Equations ofPoincar~, PMM 5, 253-262 (1941). [2] Q.K. GHORI and M.HusSAtN, Poincard's Equations for Nonholonomic Dynamical Systems,

ZAMM 53-7, 391-396 (1973). [3] Q.K. GMORI and M. HusskIN, Generalization of the Hamilton-Jacobi Theorem, J. Appl. Math.

Phys. (ZAMP), 25, 536-540 (1974). [4] E.KH.NAzmv, On the Hamilton-Jacobi Method for Nonholonomic Systems, PMM 36-6,

1108-1114 (1972). [5] L.A.PARS, A Treatise on Analytical Dynamics, Heinemann Press, London (1968). [6] H.PoINCARI~, On a New Form of the Equations o f Mechanics, C.R. Acad. Sci. 132, 369-371

(1901).

Summary

In this paper we have considered a conservative holonomic dynamical system whose position is defined by a certain number of group variables. The Hamilton-Jacobi theorem for this system has been proved by the method of direct verification.

Zusammenfassung

Fi~r konservative holonome Systeme, deren Lage durch eine gewisse Zahl yon Gruppen- variablen definiert ist, wird das Theorem yon Hamilton-Jacobi mittels direkter Verifikation bewiesen.

(Received: April 25, 1975; revised: October 1, 1975)