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 $s_$ on $N$ such that the Poisson bracket of integrals of motion reads
$\= s_\circ F$.

(iii) The matrix function $s_$ 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 $T^m$. There exists an open neighbourhood $U$ of $F$ 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 $T^\times\mathbb R^r$.

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.
* Bolsinov, A., Jovanovic, B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003) 305; .
* 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; .
* Miller, W., Jr, Post, S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001,
* Giachetta, G., Mangiarotti, L., Sardanashvily, G., ''Geometric Methods in Classical and Quantum Mechanics'' (World Scientific, Singapore, 2010) ; {{arXiv, 1303.5363.

Hamiltonian mechanics
Dynamical systems
Integrable systems