Extensions of Noether's Second Theorem: from continuous to discrete systems

TL;DR

This paper presents a straightforward proof of Noether's Second Theorem and generalizes it to both continuous and discrete systems, providing a unified approach to conservation laws and relationships in variational problems.

Contribution

It introduces a simple proof method that extends Noether's Second Theorem to discrete systems and offers practical applications to various well-known systems.

Findings

Generalized conservation laws for variational problems

Unified treatment of continuous and discrete systems

Illustrations on well-known systems

Abstract

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler--Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations.