Generalized Cartesian–Nambu Vector Fields

Abstract

Nambu mechanics is a generalization of Hamiltonian mechanics involving several Hamiltonians. Recall that Hamiltonian mechanics is based on the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms, i.e., a transformation of phase space that is volume preserving and preserves the symplectic structure of the phase space, and hence obeys Liouville’s Theorem. In 1973 Yoichiro Nambu suggested an extension of Hamiltonian dynamics, based on an N-dimensional Nambu–Poisson manifold replacing the even dimensional Poisson manifold, i.e., a manifold with a given Poisson bracket and replacing a single Hamiltonian H for N − 1 Hamiltonian H1, …, HN−1. In the canonical Hamiltonian formulation the equations of motion (Hamilton equations) are defined via the Poisson bracket.