Existence of KAM tori in degenerate Hamiltonian systems
type: Item
instance of: scholarly article
title: Existence of KAM tori in degenerate Hamiltonian systems
zbMATH Open document ID: 0813.58050
author: Chongqing Cheng
author: Yisui Sun
published in: Journal of Differential Equations
publication date: 1995-06-06T00:00:00Z
review text: The work extends the standard Kolmogorov-Arnold-Moser theorem to the perturbed systems with the Hamiltonian \(H(p,q) = N(p) + P(p,q)\), where \(p\) and \(q\) are the \(n\)-dimensional vectors of the action and angle variables, and, in contrast with the usual formulation of the theorem, it is not assumed that the mapping from the action space to the frequency space, characterized by the Jacobian matrix \(\partial N/\partial p\), is nondegenerate. In this work, only real analytical Hamiltonians are considered. The demand of the nondegenerateness is replaced by an essentially weaker condition. Namely, the following theorem is proved: the small perturbation conserves almost all the invariant tori of the unperturbed system, only slightly deforming them, provided that in the domain \(\Sigma\) of analyticity of the Hamiltonian, in which the Hamiltonian is \(2 \pi\)-periodic in the angular variables \(q\), the rank \(r\) of the mapping \(\partial N/\partial p\) (\(1 \leq r \leq n\)) is constant, and the image of any open set in \(\text{Re}(\Sigma)\) includes a curved \(C^{n + 2}\) one-dimensional submanifold. The proof of the theorem is based on analysis of distribution of the Diophantine vectors in the \(n\)-dimensional Euclidean space. By foliating the action space locally into a family of curves so that an image of each curve is a curved one-dimensional manifold, the proof is restricted to the existence of KAM tori on those curves. The fact that the image manifolds do not get flat under the action of the small perturbation allows to prove that the Newtonian iteration scheme does not lose its superconvergence, provided that the Diophantine constant is changed properly at each step of the iteration.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70F10
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 696892
zbMATH Keywords: degenerate Hamiltonian systems
zbMATH Keywords: Diophantine vectors
zbMATH Keywords: Kolmogorov-Arnold- Moser
zbMATH Keywords: perturbation
zbMATH Keywords: invariant tori
reviewed by: B. A. Malomed
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1152
OpenAlex ID: W2085560703
DOI: 10.1006/JDEQ.1994.1152
Recommended article: A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems
Recommended article: Degenerate lower-dimensional tori in Hamiltonian systems
Recommended article: A KAM theorem of degenerate infinite dimensional Hamiltonian systems. II
Recommended article: A KAM theorem of degenerate infinite dimensional Hamiltonian systems. I
Recommended article: KAM theory for lower dimensional tori of nearly integrable Hamiltonian systems
Recommended article: Invariant tori for nearly integrable Hamiltonian systems with degeneracy
Recommended article: Invariant KAM tori and global stability for Hamiltonian systems
Recommended article: Invariant hyperbolic tori for Hamiltonian systems with degeneracy
Recommended article: Q3072895
Recommended article: Q4675167
description: scientific article; zbMATH DE number 696892