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 n ≤ k 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. 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, \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.

See also

References

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