On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System

This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincaré methods. An algorithm is presented for constructing them in the form of series (fractional degrees of a small parameter), which is implemented using classical perturbation theory based on the theory of canonical transformations of Hamiltonian systems. The problem of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The results obtained are applied to the problem of subharmonic oscillations of a pendulum placed on a moving platform in a homogeneous gravitational field. The platform rotates with constant angular velocity about a vertical passing through the suspension point of the pendulum, and simultaneously executes harmonic small-amplitude oscillations along the vertical. Families of subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability is solved.