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

The problem of the orbital stability of periodic motions of a Hamiltonian s ystem 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 m ultipliers are real and have moduli equal to unity, is investigated. The st ability and instability conditions are obtained using Lyapunov's second met hod and the KAM theory. Constructive algorithms for checking these conditio ns are given. The case of a system containing a small parameter is consider ed in particular. On the technical side, the investigation rests primarily on the classical theory of perturbations of Hamiltonian systems and its mod ern modifications. The problem of the stability of the permanent rotation o f 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 ab out a fixed point are considered as applications. (C) 2001 Elsevier Science Ltd. All rights reserved.