In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system-dimensional symplectic manifold for which the following conditions hold:

.

such that the Poisson bracket.

.

, 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.

is a fiber bundle. These coordinates are the Darboux coordinates which are the Casimir functions of the coinduced Poisson structure.

The Liouville-Arnold theorem for completely integrable systems.

See also

• Integrable system

• Action-angle coordinates

• Nambu mechanics

• Laplace–Runge–Lenz vector

• Fradkin tensor

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) ; .