Investigation of the stability of periodic motions of an autonomous Hamiltonian system in a critical case

Abstract

The problem of the orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom is considered. The Hamilton function does not depend explicitly on the time and is analytic in the neighbourhood of the trajectory of the unperturbed motion. The critical case, when all the multipliers are real and have moduli equal to unity, is investigated. The stability and instability conditions are obtained using Lyapunov's second method and the KAM theory. Constructive algorithms for checking these conditions are given. The case of a system containing a small parameter is considered in particular. On the technical side, the investigation rests primarily on the classical theory of perturbations of Hamiltonian systems and its modern modifications. The problem of the stability of the permanent rotation of a heavy circular disc which is in collision with a fixed horizontal plane and the problem of the stability of the plane rotations of a rigid body about a fixed point are considered as applications.