# Superintegrable Hamiltonian system

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 $k>n$ independent integrals $F_{i}$ of motion. Their level surfaces (invariant submanifolds) form a fibered manifold $F:Z\to N=F(Z)$ over a connected open subset $N\subset \mathbb {R} ^{k}$ .

(ii) There exist smooth real functions $s_{ij}$ on $N$ such that the Poisson bracket of integrals of motion reads $\{F_{i},F_{j}\}=s_{ij}\circ F$ .

(iii) The matrix function $s_{ij}$ is of constant corank $m=2n-k$ 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 $T^{m}$ . There exists an open neighbourhood $U$ of $F$ which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates $(I_{A},p_{i},q^{i},\phi ^{A})$ , $A=1,\ldots ,m$ , $i=1,\ldots ,n-m$ such that $(\phi ^{A})$ 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 $T^{m-r}\times \mathbb {R} ^{r}$ .