Theory of tensor invariants of integrable Hamiltonian systems. I: Incompatible Poisson structures
type: Item
instance of: scholarly article
title: Theory of tensor invariants of integrable Hamiltonian systems. I: Incompatible Poisson structures
zbMATH Open document ID: 0861.58021
DOI: 10.1007/BF02099623
published in: Communications in Mathematical Physics
publication date: 1997-05-12T00:00:00Z
review text: The paper represents an interesting contribution to the theory of invariants of completely integrable Hamiltonian systems on a smooth manifold, initiated by \textit{F. Magri} [J. Math. Phys. 19, 1156-1162 (1978; Zbl 0383.35065)]. The author gives a complete classification of all invariant closed 2-forms and corresponding invariant nondegenerate Poisson structures in a neighbourhood of an invariant toroidal domain and shows that these Poisson structures are in general incompatible with the original one. Applications connected with the KAM theory, Kepler problem, celestial mechanics and the classical harmonic oscillator are presented. The concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced and related to complete integrability of a dynamical system.
reviewed by: Alexandr Vondra
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 70F15
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 934767
zbMATH Keywords: integrable Hamiltonian system
zbMATH Keywords: Poisson structure
zbMATH Keywords: dynamical compatibility
zbMATH Keywords: complete integrability
author: Oleg I. Bogoyavlenskij
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02099623
OpenAlex ID: W2041365405
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description: scientific article; zbMATH DE number 934767