Potential Conservation Laws

TL;DR

This paper characterizes potential conservation laws, showing they depend only on local variables unless induced by local laws, and extends previous results with new tools and applications in differential equations.

Contribution

It generalizes existing results on potential conservation laws, introduces new tools like weighted jet spaces, and explores extensions to gauged systems and coverings.

Findings

Potential conservation laws depend only on local variables unless induced by local laws.

Pure potential conservation laws fundamentally depend on potential variables.

The paper provides an example illustrating the applications of these theoretical results.

Abstract

We prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws. Therefore, characteristics of pure potential conservation laws have to essentially depend on potential variables. This statement provides a significant generalization of results of the recent paper by Bluman, Cheviakov and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present extensions to gauged potential systems, Abelian and general coverings and general foliated systems of differential equations. An example illustrating possible applications of proved statements is considered. A special version of the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws.