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

Peter E. Hydon, Elizabeth L. Mansfield

Published 2011-03-09Version 1

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.