Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a ${\displaystyle 2n}$-dimensional symplectic manifold for which the following conditions hold:

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

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

(iii) The matrix function ${\displaystyle s_{ij}}$ is of constant corank ${\displaystyle m=2n-k}$ on ${\displaystyle N}$.

If ${\displaystyle k=n}$, 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 ${\displaystyle F}$ is a fiber bundle in tori ${\displaystyle T^{m}}$. There exists an open neighbourhood ${\displaystyle U}$ of ${\displaystyle F}$ which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates ${\displaystyle (I_{A},p_{i},q^{i},\phi ^{A})}$, ${\displaystyle A=1,\ldots ,m}$, ${\displaystyle i=1,\ldots ,n-m}$ such that ${\displaystyle (\phi ^{A})}$ are coordinates on ${\displaystyle T^{m}}$. These coordinates are the Darboux coordinates on a symplectic manifold ${\displaystyle U}$. A Hamiltonian of a superintegrable system depends only on the action variables ${\displaystyle I_{A}}$ which are the Casimir functions of the coinduced Poisson structure on ${\displaystyle F(U)}$.

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 ${\displaystyle T^{m-r}\times \mathbb {R} ^{r}}$.