On the Inversion of Noether's Theorem in Classical Dynamical Systems

A simple and conceptually clear proof of the inverse Noether's theorem is presented, both for classical point mechanics and for classical field theory. We start from an analysis of the time dependence (of the four divergence, respectively), of an arbitrary dynamical quantity, to relate it to the variation of the action integral induced by a suitably defined infinitesimal transformation. When the dynamical quantity is chosen to be a constant of motion (a divergenceless quantity, respectively), this infinitesimal transformation is shown to act as an invariance transformation on the dynamical system.