KAM Theorems

Abstract

This chapter contains the formulations, simple consequences, and some applications of the main theorem of KAM theory. It asserts that there is a massive set of invariant tori, bearing quasiperiodic motions with suitable frequencies, which persists under small perturbations. The typical situation is a small Hamiltonian perturbation of a completely integrable system which has a foliation on invariant Lagrangian tori, as we have seen in §7. Roughly speaking, some of these tori persist and some are destroyed as one switches on a small perturbation. In the more general situation, the notion of a KAM set, which we shall derive in §10, appears to be an adequate one in describing the phenomenon in question. KAM sets exist in many familiar dynamical systems (see §§11–14) and they also persist under small perturbations (§15).