Moving Frames and Conservation Laws for Euclidean Invariant Lagrangians

TL;DR

This paper explores how the mathematical structure of differential invariants and moving frames simplifies finding extremal curves in variational problems with Euclidean symmetry, extending classical conservation law methods.

Contribution

It demonstrates that understanding the structure of differential invariants and moving frames significantly eases the process of finding extremal curves for Euclidean invariant variational problems.

Findings

Simplifies the derivation of extremal curves for SE(2) and SE(3) invariant problems.

Provides explicit formulas leveraging differential invariants and moving frames.

Enhances the application of Noether's theorem in geometric variational calculus.

Abstract

Noether's First Theorem yields conservation laws for Lagrangians 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. In recent work the authors showed the mathematical structure behind 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. In this paper we demonstrate that the knowledge of this structure considerably eases finding the extremal curves for variational problems invariant under the special Euclidean groups SE(2) and SE(3).