Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a ${\displaystyle 2n}$-dimensional symplectic manifold for which the following conditions hold:

(i) There exist ${\displaystyle k>n}$ independent integrals ${\displaystyle F_{i}}$ of motion. Their level surfaces (invariant submanifolds) form a fibered manifold ${\displaystyle F:Z\to N=F(Z)}$ over a connected open subset ${\displaystyle N\subset \mathbb {R} ^{k}}$.

(ii) There exist smooth real functions ${\displaystyle s_{ij}}$ on ${\displaystyle N}$ such that the Poisson bracket of integrals of motion reads ${\displaystyle \{F_{i},F_{j}\}=s_{ij}\circ F}$.

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

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

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 ${\displaystyle T^{m-r}\times \mathbb {R} ^{r}}$.

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. 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.