KAM Theory for secondary tori

TL;DR

This paper proves a result about the persistence of invariant tori in nearly-integrable Hamiltonian systems with real-analytic potentials, showing that most tori survive small perturbations, with the measure of destroyed tori being very small.

Contribution

It provides a rigorous proof of a previously announced theorem on the measure of invariant tori in nearly-integrable systems with general non-degenerate potentials.

Findings

Most invariant tori persist under small perturbations.

The measure of destroyed tori is bounded by a small quantity involving epsilon.

The result applies to real-analytic Hamiltonian systems with non-degenerate potentials.

Abstract

In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145 [math.DS]) the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian system with real-analytic Hamiltonian $$H(y,x)=\frac12 \sum_{i=1}^n y_i^2 +\epsilon f(x)\ ,$$ $(y,x)\in{\mathbb R}^n\times{\mathbb T}^n$ being standard action--angle variables. For "general non-degenerate" potentials $f$'s there exists $\epsilon_0,a>0$ such that, if $0<\epsilon<\epsilon_0$, then the Liouville measure of the complementary of $H$-invariant tori is smaller than $\epsilon|\log \epsilon|^a$. In this paper we provide a proof of such result.