ACTA MECHANICA SINICA (English Series), Vol.14, No.l, Feb. 1998 The Chinese Society of Theoretical and Applied Mechanics Chinese Journal of Mechanics Press, Beijing, China Allerton Press, INC., New York, U.S.A.
ISSN 0567-7718
NOTE ON THE VIRIAL THEOREM* Q. K. Ghori
( GIK Institute of Engineering Sciences & Technology, Topi, NWFP, Pakistan) Poincar6's formalism is used to develop a variant of the usual virial theorem in which the time average of the equation of motion of a certain function is expressed in terms of the generalized Poisson brackets. ABSTRACT:
K E Y W O R D S : analytical mechanics, virial theorem, Poincar6's formalism The virial theorem of Clausius deals with time-averaged quantities like the kinetic energy of a mechanical system. A statistical average of energies is all that we need for many astronomical purposes. One such example is the deduction of the presence of dark matter in groups of galaxies. In 1972, Van Kampen [1] gave a proof of the virial theorem, in analogy to Noether's theorem, in terms of special symmetry properties of a mechanical system. Unlike the familiar derivations of the theorem, his proof restricts the form of virial theorem and the class of mechanical systems to which it applies. This restriction was removed by Kobussen [2] who generalized the virial theorem to any mechanical system which admits a Hamiltonian function. In this note we develop a variant of the virial theorem in the general setting of Poincar6's formalism [3]. Without using the symmetry properties we prove the theorem by a time integration of equation of motion of a certain arbitrary function. The result of Kobussen follows as a special case of our general theorem. Let the n independent parameters xi define at any time t the position of a mechanical system and let X / b e the group of infinitesimal displacement operators of the system with commutation relations
(x,,xj)=c}xk
where i,j, k = 1, 2 , . . . , n and k is the summation index. The variation d F of an arbitrary function F(xj, t) in a real displacement of the system is determined by the formula [4]
dF = (O--~ + ~liXiF)dt
(1)
where the independent Poincar6 parameters ~i characterize the real displacement. With F = xi in (1), we can express the velocities xl in terms of the ~ts. Thus the Lagrangian L of the ~system Can be expressed in terms of the x's, ~rs and t. We define the Hamiltonian H(xj, yj, t) by the relation
H(xj, yj, t) = Yi~?i - L(xj, yj, t) Received 16 September 1997 * Recommended by Prof. JMei Fengxiang
OL Yi = O~ii
Vol.14, No.1
Q.K. Ghori: Note on the Virial Theorem
77
Then the motion of the system is determined by the canonical equations [4]
~?i = OH Oyi
y~ = - X i H + Ckiyk OH Oy~
(2)
In view of (1) and (2), the time evolution of an arbitrary function f(xj, yj) is given by
_OH
Of
~k
Of OH
or
dI d--t- = (f' H )
(3)
where (f, H ) denotes the Poisson bracket
OH Of Of OH (f, H) = X,f-~yi - X , H ~ + C]iyk Yi Oyi Oyj With fp = fp(xj(t), yj(t)),p = 1, 2, we integrate Eq.(3) with respect to the time from t, to t2 where tl --+ - o o , t2 -+ ~-oo. Thus
lim
/2
I1
=/xjOH\
,~-~-o= t2 -- tl t2--~+oo
-
\
(4)
-
Oyi /t
Yi
t
Here (K>t denotes the time average
lim
=
1
tl-+--~176 t 2 - - ~I t2--~+ao
['2 g(t)dt
AI
In the special case where f(xj, yj) is bounded for all times, the left-hand side of (4) becomes zero and we obtain
X, fOH ~ 9 since
Of OH Oy~
= -C],. Putting Yi = xl, we have
Xi
= ~
0
and all the C ~ vanish. Then (4) and (5) assume
the form obtained by Kobussen [2] lira
f2--fl :_(Of OH) _/OfOHh
tl'+--~176 t2--tl t2-~+oo and
<
~fxi
OH
-~Xi--~yi t
\OyiOxi/t
Of OH
A further specialization with f = y~xi leads to the familiar formulation of the virial theorem, namely Thus Eqs.(4) and (5) are generalizations of the virial theorem.
REFERENCES 1 2 3 4
Van Kampen NG. Reports Math Phys. 1972, 3:235 Kobussen JA. Reports Math Phys. 1976, 10:245~247 Chetaev NG. On the equations of Poincar4. PMM, 1941, 5(2): 253,,~262 Ghori QK, Hussain M. Generalization of Hamilton Jacobi Theorem. ZAMP, 1974, 25:536,,~540
