A note on micro-instability for Hamiltonian systems close to integrable

TL;DR

This paper demonstrates a form of micro-diffusion in near-integrable Hamiltonian systems, showing that small perturbations can cause action variables to drift significantly over specific timescales.

Contribution

It proves the existence of orbits with notable action drift in perturbed Hamiltonian systems under minimal generic conditions.

Findings

Existence of orbits with action drift of order sqrt(epsilon)

Drift occurs over timescale inversely proportional to sqrt(epsilon)

Results are essentially optimal within the considered setting

Abstract

In this note, we consider the dynamics associated to an epsilon-perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of "micro-diffusion": under generic assumptions on h and f , there exists an orbit of the system for which the drift of its action variables is at least of order square root of epsilon, after a time of order the inverse of square root of epsilon. The assumptions, which are essentially minimal, are that there exists a resonant point for h and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.