Physics:Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a [math]\displaystyle{ 2n }[/math]-dimensional symplectic manifold for which the following conditions hold: (i) There exist [math]\displaystyle{ k\gt n }[/math] independent integrals [math]\displaystyle{ F_i }[/math] of motion. Their level surfaces (invariant submanifolds) form a fibered manifold [math]\displaystyle{ F:Z\to N=F(Z) }[/math] over a connected open subset [math]\displaystyle{ N\subset\mathbb R^k }[/math].

(ii) There exist smooth real functions [math]\displaystyle{ s_{ij} }[/math] on [math]\displaystyle{ N }[/math] such that the Poisson bracket of integrals of motion reads [math]\displaystyle{ \{F_i,F_j\}= s_{ij}\circ F }[/math].

(iii) The matrix function [math]\displaystyle{ s_{ij} }[/math] is of constant corank [math]\displaystyle{ m=2n-k }[/math] on [math]\displaystyle{ N }[/math].

If [math]\displaystyle{ k=n }[/math], 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 [math]\displaystyle{ F }[/math] is a fiber bundle in tori [math]\displaystyle{ T^m }[/math]. There exists an open neighbourhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ F }[/math] which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates [math]\displaystyle{ (I_A,p_i,q^i, \phi^A) }[/math], [math]\displaystyle{ A=1,\ldots, m }[/math], [math]\displaystyle{ i=1,\ldots,n-m }[/math] such that [math]\displaystyle{ (\phi^A) }[/math] are coordinates on [math]\displaystyle{ T^m }[/math]. These coordinates are the Darboux coordinates on a symplectic manifold [math]\displaystyle{ U }[/math]. A Hamiltonian of a superintegrable system depends only on the action variables [math]\displaystyle{ I_A }[/math] which are the Casimir functions of the coinduced Poisson structure on [math]\displaystyle{ F(U) }[/math].

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 [math]\displaystyle{ T^{m-r}\times\mathbb R^r }[/math].

See also

• Integrable system
• Action-angle coordinates
• Nambu mechanics
• Laplace–Runge–Lenz vector

References

• Mishchenko, A., Fomenko, A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978) 113. doi:10.1007/BF01076254
• 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. doi:10.1007/s10440-005-1139-8
• 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.
• Miller, W., Jr, Post, S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001, doi:10.1088/1751-8113/46/42/423001 arXiv:1309.2694
• 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.