Noether theorems and higher derivatives

Abstract

A simple proof of Noether's first theorem involves the promotion of a constant symmetry parameter $\epsilon$ to an arbitrary function of time, the Noether charge $Q$ is then the coefficient of $\dot\epsilon$ in the variation of the action. Here we examine the validity of this proof for Lagrangian mechanics with arbitrarily-high time derivatives, in which context "higher-level" analogs of Noether's theorem can be similarly proved, and "Noetherian charges" read off from, e.g. the coefficient of $\ddot \epsilon$ in the variation of the action. While $Q=0$ implies a restricted gauge invariance, unrestricted gauge invariance requires zero Noetherian charges too. Some illustrative examples are considered and the extension to field theory discussed.