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