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 Poincaré-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 Poincaré map, also introduced by Nekhoroshev.

21 pages, no figures

References in corpus (1)

• On persistence of invariant tori and a theorem by Nekhoroshev

Cited by in corpus (14)

• The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds
• Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
• Symmetries and Integrability
• Bi-Hamiltonian partially integrable systems
• Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds
• Univalent functions and integrable systems
• Noncommutative integrability on noncompact invariant manifolds
• NC Effective Gauge Model for Multilayer FQH States
• Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system
• Systems of Hess-Appel'rot Type and Zhukovskii Property
• The Poincare'-Lyapounov-Nekhoroshev theorem for involutory systems of vector fields
• Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory
• Geometrical aspects of integrable systems
• Partially superintegrable systems on Poisson manifolds