Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions

TL;DR

This paper proves the persistence of certain lower-dimensional invariant tori in non-convex Hamiltonian systems under perturbations, using multiscale analysis and exploiting Hamiltonian symmetries.

Contribution

It demonstrates the existence of lower-dimensional invariant tori in perturbed non-convex Hamiltonian systems, extending KAM theory to this class.

Findings

Existence of at least one lower-dimensional torus with a Diophantine rotation vector after perturbation.

Application of multiscale analysis and resummation to handle divergent series.

Identification of symmetries and cancellations due to Hamiltonian structure.

Abstract

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.