The Poincaré-Lyapunov-Liouville-Arnol'd theorem
type: Item
instance of: scholarly article
title: The Poincaré-Lyapunov-Liouville-Arnol'd theorem
zbMATH Open document ID: 0847.58035
DOI: 10.1007/BF01076504
publication date: 1995-09-27T00:00:00Z
review text: The author presents the following theorem: Let \(M\) be a symplectic manifold of dimension \(2n\). Suppose that a Hamiltonian flow \(X_H\), \(H \in C^\infty (M)\), possesses \(k\) \((1 \leq k \leq n)\) integrals in involution \(H= F_1,F_2,\dots, F_k\) and that there exists a \(k\)-dimensional compact connected submanifold \(S \subset M\) invariant under the Hamiltonian flows of all the \(F_i\)'s with \(dF_1,\dots, dF_k \mid S\) linearly independent. Then \(S\) is a torus. If \(k < n\) assume in addition that the fundamental group \(\pi_1(S)\) has at least one element whose monodromy (linearized Poincaré map) spectrum does not contain \(1\). Then, in some neighborhood \(U\) of \(S\) there exists a \(2k\)-dimensional symplectic submanifold \(N\) which is a trivial fibration whose fibers are \(k\)-dimensional tori \(S_\beta = N \cap (F_1,\dots, F_k)^{-1} (\beta)\), \((F_1,\dots, F_k) (S_\beta) = \beta\), one of the latter tori coinciding with \(S\). Moreover \(N\) admits action-angle variables consistent with the above fibration, and on each fiber \(S_\beta\) the Hamiltonian flows of the \(F_i\)'s are quasi-periodic. This result provides an interpolation between the Poincaré-Lyapunov theorem \((k =1)\) and the Liouville-Arnol'd theorem \((k = n)\).
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 70G10
zbMATH DE Number: 795622
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: symplectic geometry
zbMATH Keywords: foliations
zbMATH Keywords: invariant tori
zbMATH Keywords: Poincaré-Lyapunov theorem
zbMATH Keywords: Liouville-Arnol'd theorem
author: N. N. Nekhoroshev
reviewed by: Serge Parmentier
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf01076504
OpenAlex ID: W2074006882
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 795622