Superintegrable Hamiltonian system
(i) There exist independent integrals of motion. Their level surfaces (invariant submanifolds) form a fibered manifold over a connected open subset .
(ii) There exist smooth real functions on such that the Poisson bracket of integrals of motion reads .
(iii) The matrix function is of constant corank on .
If , this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows.
Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold is a fiber bundle in tori . Given its fiber , there exists an open neighbourhood of which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates , , such that are coordinates on . These coordinates are the Darboux coordinates on a symplectic manifold . A Hamiltonian of a superintegrable system depends only on the action variables which are the Casimir functions of the coinduced Poisson structure on .
The Liouville-Arnold theorem for completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder .
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