Classical Nambu brackets in higher dimensions

TL;DR

This paper explores conditions under which higher-dimensional Hamiltonian systems can possess n-linear Nambu brackets, revealing that for n=3, such brackets exist only in systems with effectively one degree of freedom, and generalizing to higher n.

Contribution

It establishes necessary and sufficient conditions for the existence of n-linear Nambu brackets in higher-dimensional systems, extending the canonical Nambu bracket to broader contexts.

Findings

For n=3, N=K+2, systems have effectively one degree of freedom.

The canonical Nambu bracket is the unique local solution up to variable change.

Results generalize to n-linear brackets with n ≥ 4.

Abstract

We consider n-linear Nambu brackets in dimension N higher than n. Starting from a Hamiltonian system with a Poisson bracket and K Casimir invariants defined in the phase space of dimension N = K+2M, where M is the number of effective degrees of freedom, we investigate a necessary and sufficient condition for this system to possess n-linear Nambu brackets. For the case of n = 3, by looking for the possible solutions to the fundamental identity, the condition is found to be N = K+2, i.e., the system should have effectively one degree of freedom. Locally, it is shown that there is only one fundamental solution, up to a local change of variables, and this solution is the canonical Nambu bracket, generated by Levi-Civita tensors. These results generalize to the case of n($\ge$ 4)-linear Nambu brackets.