A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems
type: Item
instance of: scholarly article
title: A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems
zbMATH Open document ID: 0977.34051
DOI: 10.1016/S0393-0440(99)00025-X
author: Juan-Pablo Ortega
author: T. S. Ratiu
published in: Journal of Geometry and Physics
publication date: 2002-01-07T00:00:00Z
review text: The problem of nonlinear stability of relative equilibria of the symmetric Hamiltonian systems when the dimension of the space are bigger than two is considered. It is known that Dirichlet's theorem has been adapted already to the study of the stability of relative equilibria (periodic orbits or relative periodic orbits). In that situations it received the names of Arnold, energy-Casimir, enery-momentum method or others. All these techniques are based on the definiteness of the second variation of an augmented energy function (which in infinite dimensions needs to be replaced with certain convexity estimates) and, as in the case of Dirichlet's theorem, they provide only sufficient stability conditions. The classical Dirichlet criterion is generalized and Dirichlet's like sufficient conditions for the stability of periodic orbits and relative periodic orbits are formulated in the article. For terminological convenience and to differentiate these results from the situations dealing with equilibria and relative equilibria the authors call their conditions the energy-integrals and symmetric energy-integrals methods, respectively. Some application of the results to several elementary examples are given. A short review on the known results is given.
reviewed by: Sergei Georgievich Zhuravlev
Mathematics Subject Classification ID: 34D35
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37J25
Mathematics Subject Classification ID: 70H14
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 1396337
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: relative equilibria
zbMATH Keywords: nonlinear stability
zbMATH Keywords: Dirichlet criterion
zbMATH Keywords: sufficient conditions of stability
MaRDI profile type: Publication
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full work available at URL: s0393-0440(99)00025-x
OpenAlex ID: W2016871068
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description: scientific article; zbMATH DE number 1396337