Classical Nambu brackets in higher dimensions

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–Cività tensors. These results generalize to the case of n(≥4)-linear Nambu brackets.