Introduction to KAM Theory

2024/2025

Academic Year

ENG

Instruction in English

3

ECTS credits

Delivered at:

Faculty of Mathematics

Course type:

Optional course (faculty)

When:

3, 4 module

Instructor

Glutsyuk, Alexey

Course Syllabus

Abstract

Hamilton equations are one of the most fundamental class of differential equations describing law of movements of mechanical systems and many physical laws. Their flows preserve symplectic structure and hence, the corresponding volume form. The simplest class of Hamiltonian systems are integrable Hamiltonian systems on compact symplectic manifolds. In this case Arnold -- Liouville Theorem states that the manifold is fibered by invariant half-dimensional tori along which the movement is quasi-periodic: its trajectories follow integral curves of linear flow on torus. The KAM theory created by A.N.Kolmogorov, V.I.Arnold and J.Moser in late 1950-ths - early 1960-ths deals with perturbations of the so-called non-degenerate integrable Hamiltonian systems: namely, those of them, where the frequency vector of quasi-periodic movement along an invariant torus has non-degenerate derivative in the transversal parameter. The main KAM theorem states that for every small perturbation most of invariant tori survive: the smaller is the perturbation parameter, the more percentage (in the sense of Lebesgue measure) of survived tori. The KAM theory is used in many domains of mathematics, mechanics and physics. First of all, in symplectic dynamics and in celestial mechanics. And also in the theory of billiards. The goal of the cours is to present the KAM theory with a proof of the KAM theorem and its different versions, and to discuss its applications and related questions. Including the case of perturbations of integrable twist symplectomorphisms of planar cylinder (persistance of most of invariant circles) and billiard theory (Lazutkin's Theorem on existence of Cantor family (of positive measure) of caustics: curves whose tangent lines are reflected again to their tangent lines). We will also discuss global dynamics of the perturbed system; first of all, for perturbations of integrable twist symplectomorphism of cylinder.