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

References

Mishchenko, A., Fomenko,A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978) 113.
Bolsinov, A., Jovanovic, B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003) 305; arXiv:math-ph/0109031.
Fasso, F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87(2005) 93.
Fiorani, E., Sardanashvily, G., Global action-angle coordinates for completely integrable systems with non-compact invariant manifolds, J. Math. Phys. 48 (2007) 032901; arXiv:math/0610790.
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Methods in Classical and Quantum Mechanics (World Scientific, Singapore, 2010) ISBN 978-981-4313-72-8; arXiv: 1303.5363.

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Superintegrable Hamiltonian system

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References