Journal of Applied Mathematics and Mechanics (about journal) Journal of Applied
Mathematics and Mechanics

Russian Academy of Sciences
in January 1936
(Translated from 1958)
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IssuesArchive of Issues2015-2pp.122-131

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V.N. Tkhai, "Periodic motions of a perturbed reversible mechanical system," J. Appl. Math. Mech. 79 (2), 122-131 (2015)
Year 2015 Volume 79 Issue 2 Pages 122-131
DOI 10.1016/j.jappmathmech.2015.07.002
Title Periodic motions of a perturbed reversible mechanical system
Author(s) V.N. Tkhai (Moscow, Russia,
Abstract Periodic motions of a reversible mechanical system under perturbations of general form that do not preserve system's reversibility are investigated. The conditions for the existence of periodic motions of a perturbed system that turn at a zero value of a small parameter into symmetrical periodic motions not degenerating into equilibriua are found. Both autonomous and periodic perturbations are considered. Amplitude equations, whose simple roots correspond to periodic solutions of the perturbed system, are derived for the systems investigated. As a result, cycles are found for the autonomous system, and isolated periodic motions are found for the periodic system. Both a separate system and a model containing coupled subsystems are investigated. The characteristic exponents of the cycles are calculated. It is proved that any oscillation of a system that can be described by Lagrange's equations with positional forces is symmetrical and belongs to a family. It is also deduced that for the realization of a cycle or, in the case of non-autonomous forces, an isolated periodic motion, velocity-dependent forces of definite structure are needed. The results obtained are applied to the problem of the oscillations of a satellite under the action of gravitational and aerodynamic torques. The existence of asymmetrical isolated oscillations in a low-eccentricity elliptical orbit is established.
Received 10 January 2014
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