Superintegrable Hamiltonian system

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

Their level surfaces (invariant submanifolds) form a fibered manifold

over a connected open subset

(ii) There exist smooth real functions

such that the Poisson bracket of integrals of motion reads

(iii) The matrix function

is of constant corank

, 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

which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates

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