On globally invariant Euler--Lagrange equations for curves

TL;DR

This paper develops methods for computing globally invariant Euler-Lagrange equations for curves under various symmetry groups, extending previous work to higher dimensions and different geometries with practical applications.

Contribution

It introduces a global algebraic approach to invariant Euler-Lagrange equations, extending computations to four-dimensional Minkowski space and other geometries.

Findings

Derived formulas for invariant Euler-Lagrange equations in Minkowski spacetime.

Connected invariant equations with applications in projective and conformal geometries.

Discussed relations with previous local methods and foundational aspects.

Abstract

Invariant Lagrangians yield invariant Euler-Lagrange equations, and it was discussed in the literature how to compute those using various local methods. The focus of this paper is on global algebraic differential invariants. In this case the computation can be modified in several aspects. We will discuss relations with previous approaches and some foundational aspects. The theory of invariant Euler-Lagrange equations was applied to curves with respect to the motion group in the Euclidean plane and space. We expand those computations to the next dimension four (Minkowski spacetime), which already exhibits computational challenges. We also provide formulas for other examples, namely the projective and conformal (M\"obius) groups and relate to some recent applications.