KAM Theory. Part I. Group actions and the KAM problem

TL;DR

This paper introduces KAM theory by exploring symplectic geometry, invariant tori, and group actions, setting the foundation for understanding the persistence of quasi-periodic motions in Hamiltonian systems.

Contribution

It generalizes Kolmogorov's invariant torus theorem to broader contexts involving Poisson algebras and invariant Lagrangian varieties, and discusses the iteration method with finite-dimensional analogs.

Findings

Review of Darboux-Weinstein theorems and action-angle coordinates

Generalization of Kolmogorov's invariant torus theorem

Explanation of the iteration method with finite-dimensional analogs

Abstract

This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II.