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

TL;DR

This paper establishes a general algebraic framework for superintegrable systems in arbitrary dimensions, enabling the construction of multi-Hamiltonian and Nambu structures with applications to well-known integrable models.

Contribution

It provides a universal algebraic condition for functional independence of constants of motion and constructs compatible multi-Hamiltonian and Nambu structures for superintegrable systems.

Findings

Derived algebraic condition for independence of constants of motion.

Constructed normalized Nambu brackets for Hamiltonian evolution.

Applied framework to Calogero-Moser, magnetic field, and Smorodinsky-Winternitz systems.

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.