Quasi-periodic motions on symplectic tori

TL;DR

This paper extends the KAM theorem to establish the stability of quasi-periodic motions on symplectic tori, broadening the understanding of invariant structures in Hamiltonian systems.

Contribution

It generalizes Herman's result to the case of quasi-periodic motions on symplectic tori where the dimension equals twice the degrees of freedom.

Findings

Proves stability of quasi-periodic motions on symplectic tori

Extends KAM theory to new geometric settings

Provides a framework for analyzing invariant tori in Hamiltonian systems

Abstract

The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.