Lie algebras of conservation laws of variational partial differential equations

TL;DR

This paper extends Noether's theorem to variational PDEs, establishing a geometric correspondence between symmetries and conservation laws via an explicit linear map, under regularity conditions.

Contribution

It provides a coordinate-free, geometric proof of the bijective correspondence between symmetries and conservation laws for Euler-Lagrange equations, using an explicit linear map.

Findings

Establishes a one-to-one correspondence between symmetries and conservation laws.

Introduces an explicit linear map f_A linking symmetries to conserved quantities.

Proves that any conservation law is equivalent to one generated by the map f_A.

Abstract

We establish a version of the first Noether Theorem, according to which the (equivalence classes of) conserved quantities of given Euler-Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action, from which the Euler-Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such correspondence between symmetries and conservation laws is built on an explicit linear map f_A from the vector fields satisfying (a) and (b) into the conserved differential operators, and not into their divergences as it occurs in other proofs of Noether Theorem. This map f_A is not new: it is the map determined by contracting symmetries with a form of Poincare'-Cartan type A and it is essentially the same considered for instance in a paper by Kupershmidt. There it was shown that f_A determines a bijection between symmetries and conservation laws in a special form. Here we show that, if appropriate regularity assumptions are satisfied, any conservation law is equivalent to one that belongs to the image of f_A, proving that the corresponding induced map F_A between equivalence classes of symmetries and equivalence classes of conservation laws is actually a bijection. All results are given coordinate-free formulations and rely just on basic differential geometric properties of finite-dimensional manifolds.