On Moving Frames and Noether's Conservation Laws

TL;DR

This paper explores the mathematical structure of conservation laws and Euler-Lagrange systems using moving frames and differential invariants, enabling easier integration of equations and generalizing previous results.

Contribution

It introduces a framework based on moving frames and invariant calculus to analyze conservation laws, extending previous work to high-dimensional problems and specific group actions.

Findings

Enhanced understanding of the structure of conservation laws

Simplified integration of Euler-Lagrange equations in examples

Classification of SL(2) actions in the plane

Abstract

Noether's Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler-Lagrange system and the set of conservation laws, in terms of the differential invariants of the group action and a moving frame. For the examples we demonstrate, knowledge of this structure allows the Euler-Lagrange equations to be integrated with relative ease. Our methods take advantage of recent advances in the theory of moving frames by Fels and Olver, and in the symbolic invariant calculus by Hubert. The results here generalise those appearing in Kogan and Olver [1] and in Mansfield [2]. In particular, we show results for high dimensional problems and classify those for the three inequivalent SL(2) actions in the plane.