Liouville Integrable Systems

  • Camille Laurent-Gengoux
  • Anne Pichereau
  • Pol Vanhaecke
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 347)


The present chapter deals with the main application of Poisson structures: the theory of integrable Hamiltonian systems. We give the basic definitions and properties of functions in involution and of the momentum map, associated to them. We also give several constructions of functions in involution: Poisson’s theorem, the Hamiltonian form of Noether’s theorem, bi-Hamiltonian vector fields, Thimm’s method, Lax equations and the Adler–Kostant–Symes theorem. Liouville’s theorem and the action-angle theorem, which are classically known for integrable systems on symplectic manifolds, are presented here in the context of general Poisson manifolds.


Poisson Bracket Poisson Structure Hamiltonian Vector Poisson Manifold Liouville Theorem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Camille Laurent-Gengoux
    • 1
  • Anne Pichereau
    • 2
  • Pol Vanhaecke
    • 3
  1. 1.CNRS UMR 7122, Laboratoire de MathématiquesUniversité de LorraineMetzFrance
  2. 2.CNRS UMR 5208, Institut Camille JordanUniversité Jean MonnetSaint-EtienneFrance
  3. 3.CNRS UMR 7348, Lab. Mathématiques et ApplicationsUniversité de PoitiersFuturoscope ChasseneuilFrance

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