The Poincaré–Lyapounov–Nekhoroshev theorem

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 ≤ k ≤ 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 Poincaré-Lyapounov theorem (corresponding to k = 1) and the LiouvilleArnold one (corresponding to k = n), and interpolates between them. The crucial tool for the proof is a generalization of the Poincaré map, also introduced by Nekhoroshev.