For a Given, Time Independent, System of Equations of Motion, Any Non-Trivial Constant of Motion is Locally a Hamiltonian

Abstract

Given time-independent system of equations of motion and given a local, non-trivial constant of motion for these equations, it is shown that there exists a locally defined system of dynamical brackets, such that this given constant of motion, used together with these dynamical brackets, becomes a local Hamiltonian for the given system of equations of motion.