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In mathematics, a '''superintegrable Hamiltonian system''' is a [[Hamiltonian system]] on a  2''n''-dimensional  [[symplectic manifold]] for which the following conditions hold:
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(i) There exist ''n''&nbsp;≤&nbsp;''k'' independent integrals ''F''<sub>&nbsp;''i''</sub> of motion. Their level surfaces (invariant submanifolds) form a fibered manifold <math>F:Z\to N=F(Z)</math> over a connected open subset <math>N\subset\mathbb R^k</math>.
 
(ii) There exist smooth real functions <math>s_{ij}</math> on <math>N</math> such that the [[Poisson manifold|Poisson bracket]] of integrals of motion reads
<math>\{F_i,F_j\}= s_{ij}\circ F</math>.
 
(iii) The matrix function  <math>s_{ij}</math> is of constant corank <math>m=2n-k</math> on <math>N</math>.
 
If <math>k=n</math>, this is the case of a [[integrable system|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 <math>F</math> is a [[fiber bundle]]
in tori <math>T^m</math>. Given its fiber <math>M</math>, there exists an open neighbourhood <math>U</math>  of <math>M</math> which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates <math>(I_A,p_i,q^i, \phi^A)</math>,
<math>A=1,\ldots, m</math>, <math>i=1,\ldots,n-m</math> such that <math>(\phi^A)</math> are coordinates on <math>T^m</math>.  These coordinates are the [[Darboux's theorem|Darboux coordinates]] on a symplectic manifold <math>U</math>. A Hamiltonian of a superintegrable system depends only on the action variables <math>I_A</math> which are the Casimir functions of the coinduced [[Poisson manifold|Poisson structure]] on <math>F(U)</math>.
 
The  Liouville-Arnold theorem for [[Integrable system|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 <math>T^{m-r}\times\mathbb R^r</math>.
 
== See also ==
 
*[[Integrable system]]
*[[Action-angle coordinates]]
 
== 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., [[Gennadi Sardanashvily|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., [[Gennadi Sardanashvily|Sardanashvily, G.]], ''Geometric Methods in Classical and Quantum Mechanics'' (World Scientific, Singapore, 2010) ISBN 978-981-4313-72-8; [http://xxx.lanl.gov/abs/1303.5363 arXiv: 1303.5363].
 
 
 
[[Category:Hamiltonian mechanics]]
[[Category:Dynamical systems]]

Latest revision as of 09:10, 11 January 2015

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