In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2n-dimensional symplectic manifold for which the following conditions hold:
(i) There exist n ≤ k independent integrals F i of motion. Their level surfaces (invariant submanifolds) form a fibered manifold F : Z → N = F ( Z ) over a connected open subset N ⊂ R k .
(ii) There exist smooth real functions s i j on N such that the Poisson bracket of integrals of motion reads { F i , F j } = s i j ∘ F .
(iii) The matrix function s i j is of constant corank m = 2 n − k on N .
If k = n , 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 F is a fiber bundle in tori T m . Given its fiber M , there exists an open neighbourhood U of M which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates ( I A , p i , q i , ϕ A ) , A = 1 , … , m , i = 1 , … , n − m such that ( ϕ A ) are coordinates on T m . These coordinates are the Darboux coordinates on a symplectic manifold U . A Hamiltonian of a superintegrable system depends only on the action variables I A which are the Casimir functions of the coinduced Poisson structure on F ( U ) .
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 T m − r × R r .
