Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems
type: Item
instance of: scholarly article
title: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems
zbMATH Open document ID: 0861.58030
DOI: 10.1007/BF02099537
published in: Communications in Mathematical Physics
publication date: 1996-07-18T00:00:00Z
review text: The starting point for this analysis is a perturbation of an integrable Hamiltonian system, whose frequencies at a point \(p_0\) satisfy a resonance condition. The torus \(p=p_0\) of the unperturbed system being foliated into a family of \((n-1)\)-dimensional submanifolds, the idea is to investigate whether some of these submanifolds survive and still carry a quasi-periodic flow. The main theorem of the paper states that such lower dimensional tori indeed survive small perturbations under the following assumptions: the perturbed Hamiltonian is real analytic in some neighbourhood; the unperturbed part is convex; the frequency of the unperturbed flow on the \((n-1)\)-dimensional submanifold satisfy a KAM-type inequality. The surviving tori are generically of hyperbolic type. Proving this theorem, as usual in this area, is a very long and technical matter.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C55
Mathematics Subject Classification ID: 34C30
zbMATH DE Number: 906584
zbMATH Keywords: perturbed Hamiltonian systems
zbMATH Keywords: KAM-theory
zbMATH Keywords: lower dimensional invariant tori
author: Chongqing Cheng
reviewed by: W. Sarlet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 906584