Superintegrable Hamiltonian system
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold:
There exist independent integrals of motion. Their level surfaces form a fibered manifold over a connected open subset.
There exist smooth real functions on such that the Poisson bracket of integrals of motion reads
.
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. There exists an open neighbourhood of which is a trivial fiber bundle provided with the bundle 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.
