The Poincare'-Lyapounov-Nekhoroshev theorem

TL;DR

This paper provides a detailed geometric proof of Nekhoroshev's theorem on the persistence of invariant tori in Hamiltonian systems with multiple constants of motion, generalizing classical results like Poincaré-Lyapounov and Liouville-Arnold.

Contribution

It offers a comprehensive geometric proof of Nekhoroshev's theorem, extending classical theorems to systems with multiple constants of motion and introducing a generalized Poincaré map.

Findings

Persistence of k-dimensional invariant tori under nondegeneracy conditions

Existence of local partial action-angle coordinates

Generalization of the Poincaré map for Hamiltonian systems

Abstract

We give a detailed and mainly geometric proof of a theorem by N.N. Nekhoroshev for hamiltonian systems in $n$ degrees of freedom with $k$ constants of motion in involution, where $1 \le k \le n$. This states persistence of $k$-dimensional invariant tori, and local existence of partial action-angle coordinates, under suitable nondegeneracy conditions. Thus it admits as special cases the Poincar\'e-Lyapounov theorem (corresponding to $k=1$) and the Liouville-Arnold one (corresponding to $k = n$), and interpolates between them. The crucial tool for the proof is a generalization of the Poincar\'e map, also introduced by Nekhoroshev.