Superintegrable Hamiltonian system
Introduction In mathematics, a '''superintegrable Hamiltonian system''' is a Hamiltonian system on a -dimensional  symplectic manifold for which the following conditions hold:(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 bundlein tori . 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 .
See also *Integrable system*Action-angle coordinates*Nambu mechanics*Laplace–Runge–Lenz vector
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