Superintegrable Hamiltonian system

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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

  • 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 applications, 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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