Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians

TL;DR

This paper extends Nekhoroshev stability estimates to finitely differentiable quasi-convex Hamiltonians, showing polynomial stability where exponential stability is not possible, using a Gevrey regularity approach.

Contribution

It introduces polynomial stability estimates for finitely differentiable quasi-convex Hamiltonians, adapting methods from Gevrey regularity to less smooth systems.

Findings

Exponential stability does not hold for finitely differentiable Hamiltonians.

Polynomial stability estimates are established for such systems.

The approach adapts Gevrey regularity methods to finite differentiability cases.

Abstract

A major result concerning perturbations of integrable Hamiltonian systems is given by Nekhoroshev estimates, which ensures exponential stability of all solutions provided the system is analytic and the integrable Hamiltonian not too degenerate. In the particular but important case where the latter is quasi-convex, these exponential estimates have been generalized by Marco and Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by Lochak in the analytic case. In this paper, using the same approach we will investigate the situation where the Hamiltonian is assumed to be only finitely differentiable, it is known that exponential stability does not hold but nevertheless we will prove estimates of polynomial stability.