Liouville Integrable Systems
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.
KeywordsPoisson Bracket Poisson Structure Hamiltonian Vector Poisson Manifold Liouville Theorem
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