Higher order conservation laws and a higher order Noether's theorem

Abstract

Consider a general variational problem of a functional whose domain of definition consists of integral manifolds of an exterior differential system. In particular, this induces classical variational problems with constraints. With the assumption of existence of enough admissable variations the Euler-Lagrange equations associated to this problem are obtained. By studying a spectral sequence associated to the infinite prolongation of them, we extend the classical notion of infinitesimal Noether symmetries to what we shall call the "higher order Noether symmetries," and a higher order Noether's theorem identifying the higher order conservation laws and the higher order Noether symmetries is obtained. These in turn are isomorphic to the solution space of certain linear differential operator. From these we also get a systematic method of computing the higher order conservation laws of certain determined PDE systems. © 1987.