The rotational motions of mechanical systems

Autonomous systems or systems that are periodic with respect to the indepen dent variable t, specified on R-l x T-n (T-n is a torus of dimension n) are considered. In this paper 2 pi k-periodic rotational motions (including os cillatory motions), closed on R-l x T-n (in a time Delta(t) = 2 pi k, k is an element of N, for a system that is 2 pi-periodic in t) are investigated. It is shown that for such motions a theory analogous to the theory for osc illatory motions holds. In particular, Poincare's theorem on the presence o f a zero characteristic exponent in an autonomous system, the Andronov-Vitt theorem on the stability of rotational periodic motion of an autonomous sy stem, and the theory of the continuation of rotational periodic motion with respect to a small parameter hold. The necessary and sufficient conditions for periodic rotational motion to exist are given for a reversible system, and a method is proposed for constructing all such motions. A detailed inv estigation is made of periodic rotational motions of a system, close to a c onservative system with one degree of freedom. It is shown, in particular, that steady motions of an average system correspond in Volosov's method to exact periodic rotational motions. All (2 pi k/\m\) periodic rotational mot ions of a conservative system (k is an element of N, m is an element of Z\{ 0}) are conserved (in the sense of continuation with respect to a parameter ) when small reversible perturbations, 2 pi-periodic with respect to t, act on it. It is show, in the problem of the motion of a satellite in the plan e of the elliptic orbit under gravitational forces (the Beletskii problem), that additional perturbing factors have no effect on the qualitative concl usions regarding the existence of periodic rotational and oscillatory motio ns or on the stability of such motions. (C) 1999 Elsevier Science Ltd. All rights reserved.