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

Abstract

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.

arXiv comment20 pages, 1 table (submitted for publication)

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Cited by in corpus (6)

• Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds
• Bi-Hamiltonian Structures of Chaotic Dynamical Systems in 3D
• Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials
• Momentum Map and Action-Angle Variables for Nambu Dynamics
• Nambu brackets with constraint functionals
• Nonlinear Coherent States of the Fokas-Lagerstrom Potential