On the measure of Lagrangian invariant tori in nearly--integrable mechanical systems (draft)

TL;DR

This paper analyzes the measure of invariant tori in nearly-integrable Hamiltonian systems, showing that for generic potentials and small perturbations, the measure of non-invariant regions is very small, bounded by a term involving epsilon and its logarithm.

Contribution

It provides an estimate on the measure of the complement of invariant tori in real-analytic nearly-integrable systems, extending understanding of stability regions in Hamiltonian dynamics.

Findings

The measure of non-invariant tori is less than epsilon times a logarithmic factor for small epsilon.

The results hold for generic potentials in real-analytic settings.

Invariant tori occupy most of the phase space in the considered systems.

Abstract

Consider a real--analytic nearly--integrable mechanical system with potential $f$, namely, a Hamiltonian system, having a real-analytic Hamiltonian $$ H(y,x)=\frac12 | y |^2 +\e f(x)\ , $$ $y,x$ being $n$--dimensional standard action--angle variables (and $|\cdot|$ the Euclidean norm). Then, for "general" potentials $f$'s and $\e$ small enough, the Liouville measure of the complementary of invariant tori is smaller than $\e|\ln \e|^a$ (for a suitable $a>0$).