Stability in bihamiltonian systems and multidimensional rigid body

Research output: Contribution to journalArticle

7 Scopus citations

Abstract

The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is "simpler". Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?In this paper, the problem of stability of equilibria in bihamiltonian systems is considered and it is shown that the conditions for nonlinear stability can be expressed in algebraic terms of linearization of the underlying Poisson pencil. This is used to study stability of stationary rotations of a free multidimensional rigid body.

Original languageEnglish (US)
Pages (from-to)2414-2423
Number of pages10
JournalJournal of Geometry and Physics
Volume62
Issue number12
DOIs
StatePublished - Dec 1 2012
Externally publishedYes

Keywords

  • Bihamiltonian systems
  • Energy-Casimir method
  • Multidimensional rigid body
  • Stability

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

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