Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension

Details Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension Superintegrable Systems, Multi-Hamiltonian Structures and Nambu Mechanics in an Arbitrary Dimension

2002-12-25

arxiv.org/abs/math-ph/0212070v1

Int.J.Mod.Phys. A19 (2004) 393-410

Physics

Mathematical Physics

20 pages, 1 table (submitted for publication)

Scientific paper

10.1142/S0217751X04017112

A general algebraic condition for the functional independence of 2n-1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero-Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodinsky-Winternitz potentials are considered as illustrative applications and their symmetry algebras as well as their Nambu formulations and alternative Poisson structures are presented.

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