Kolmogorov stability, the impossibility of Fermi acceleration and the existence of periodic solutions in some Hamiltonian-type systems

Abstract

In certain special systems, pertaining to various branches of mechanics and physics, there is no explicitly occurring small parameter, but a suitable change of variables will artificially introduce a small parameter into the equations of motion. One can then use the Kolmogorov-Arnol'd-Moser Theorem to prove the existence of invariant (Kolmogorov) tori, which, moreover, fill the whole of phase space except for a set of finite measure (the measure of the whole phase space is infinite). Bounds are obtained for the deformation of invariant tori and the relative measure of the Kolmogorov set. It follows from Poincaré's Geometric Theorem that in all problems under consideration there exist infinitely many periodic solutions. The results of numerical experiments in connection with Ulam's problem receive their first comprehensive, rigorous justification, and analogous propositions are derived for other situations. The case of pendulum-type systems is then considered in this context by treating the reciprocal of the momentum as the small parameter. This device was first used in connection with the equations of mechanics (when momentum = angular velocity) to determine the asymptotic behaviour of fast rotations [1]. In the case of Ulam's problem a small parameter reciprocal to the particle velocity was introduced in [2]. This paper generalizes and develops results of [3] which follows from Theorem 1 as a special case.