A simple proof of the invariant torus theorem

TL;DR

This paper presents a straightforward proof of Kolmogorov's invariant torus theorem in Hamiltonian systems, utilizing an inverse function theorem and interpolation inequalities for clarity and simplicity.

Contribution

It offers a simplified, accessible proof of the invariant torus theorem by reducing it to a well-posed inversion problem and applying an inverse function theorem in the analytic setting.

Findings

Simplified proof of Kolmogorov's theorem

Reduction to a well-posed inversion problem

Application of inverse function theorem with interpolation inequalities

Abstract

We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency obstruction from one side of the conjugacy to another. Then the proof consists in applying a simple, well suited, inverse function theorem in the analytic category, which itself relies on the Newton algorithm and on interpolation inequalities. A comparison with other proofs is included in appendix.