Superintegrable Hamiltonian system explained

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a

2 n -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 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 =2 n - 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 bundlein 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,} ^{i,q} \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} x R ^{r} .

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) ; .