The Poincare'-Lyapounov-Nekhoroshev theorem

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.