The rotational motions of mechanical systems

Abstract

Autonomous systems or systems that are periodic with respect to the independent variable t, specified on r l × t n ( t n is a torus of dimension n), are considered. In this paper 2 πk-periodic rotational motions (including oscillatory motions), closed on r l × t n ( in a time Δ t = 2πk, k ϵ n , for a system that is 2π-periodic in t) are investigated. It is shown that for such motions a theory analogous to the theory for oscillatory motions holds. In particular, Poincaré's theorem on the presence of a zero characteristic exponent in an autonomous system, the Andronov-Vitt theorem on the stability of rotational periodic motion of an autonomous system, 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 investigation is made of periodic rotational motions of a system, close to a conservative 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 πk/| m|) periodic rotational motions of a conservative system k ϵ n, m ϵ z/0 ) are conserved (in the sense of continuation with respect to a parameter) when small reversible perturbations, 2π-periodic with respect to t, act on it. It is show, in the problem of the motion of a satellite in the plane of the elliptic orbit under gravitational forces (the Beletskii problem), that additional perturbing factors have no effect on the qualitative conclusions regarding the existence of periodic rotational and oscillatory motions or on the stability of such motions.