Noether theorems and higher derivatives

TL;DR

This paper extends Noether's theorem to Lagrangian systems with higher derivatives, demonstrating how gauge invariances relate to Noetherian charges and providing examples and field theory extensions.

Contribution

It provides a generalized proof of Noether's theorem for higher-derivative Lagrangian mechanics and explores the relationship between gauge invariance and Noetherian charges.

Findings

Higher derivatives lead to 'Noetherian charges' from coefficients like dot psilon.

Unrestricted gauge invariance requires zero Noetherian charges.

Examples illustrate the extended theorem and its application to field theory.

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.