Nambu dynamics and its noncanonical Hamiltonian representation in many degrees of freedom systems

TL;DR

This paper explores how Nambu dynamics can be represented as noncanonical Hamiltonian systems even when the fundamental identity is violated, with a focus on systems of many degrees of freedom and an example involving coupled oscillators.

Contribution

It demonstrates that the Jacobi identity can hold in noncanonical Hamiltonian representations of Nambu dynamics despite violations of the fundamental identity.

Findings

The Jacobi identity can be satisfied even when the fundamental identity is violated.

Noncanonical Hamiltonian structures can represent Nambu dynamics in many degrees of freedom.

An example with coupled oscillators illustrates the theoretical findings.

Abstract

Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example, we evaluate these identities for a semiclassical system of two coupled oscillators.