2021/09/28 by Chandre, Cristel, Horikoshi, Atsushi
#Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2109.13663
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(≥ 4)-linear Nambu brackets.