The development of dynamics in the 20thcentury. Kolmogorov - Arnold - Moser (KAM)theory

Abstract

This article discusses the highlights of the creation of one of the greatest scientific achievements of the 20th century - the theory of the Kolmogorov-Arnold-Moser (KAM) and completes the picture with some previously unnoticed strokes. The focus is on A.N. Kolmogorov, who played a key role in the creation of the theory. А. Poincare has shown that in the vast majority of cases dynamical systems are non-integrable. The perturbation theory is among the principal methods for solving problems of dynamics and Poincaré brought to the forefront the significance of the disturbance of an integrable Hamiltonian system as a "fundamental problem" of dynamics. Kolmogorov’s main result in this direction can be formulated as follows: for the majority of the initial conditions and the non-degeneracy of the unperturbed motion for a sufficiently small perturbation the most non-resonant tori are only being deformed, keeping to itself the trajectory of conditionally periodic motions with constant frequencies. Resonant tori are being destroyed under the influence of the disturbance, and the trajectories become stochastic. Kolmogorov limited himself to establishing all the key components of the solution of this problem and did not complete the proof of the results. This was done a few years after the appearance of Kolmogorov’s works by V.I. Arnold and J. Moser.