Introduction to the KAM Theory

Abstract

The Kolmogorov–Arnold–Moser theory showed that quasi-periodic motions are generic in Hamiltonian systems. Moreover, they usually form a set of a positive measure in the phase space. This changed considerably the generally accepted idea of the dynamics in Hamiltonian systems close to integrable. Earlier such systems were supposed to be as a rule ergodic on compact energy levels (A dynamical system is called ergodic with respect to an invariant probability measure on the phase space if the measure of any invariant set equals zero or one.). In the present chapter we discuss basic facts and ideas of the KAM theory and prove one of the simplest theorems of this type.