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On globally invariant Euler--Lagrange equations for curves

Boris Kruglikov, Eivind Schneider, Wijnand Steneker

TL;DR

This work develops a comprehensive global invariant calculus for curves, enabling Euler-Lagrange equations to be written purely in terms of differential invariants under Lie group actions. By constructing an invariant coframe and leveraging both cross-sections and moving frames, it derives an invariant EL operator ${\cal E}_{\text{inv}}$ via matrices $\mathcal{A},\mathcal{B}$ and a relative invariant $W$, applicable across Euclidean, Möbius, projective, and conformal geometries, including 4D Minkowski. The authors demonstrate both local and global invariant EL formulations for a variety of geometries, providing explicit formulas for Lagrangians such as $L=1$, $L=\kappa^2 ds$, $L=τ ds$, and others, and analyze singular extremals where the relative invariant $W$ degenerates (e.g., conformal geodesics). They also discuss computational challenges and propose a cross-section–based approach that complements moving frames, highlighting practical strategies for symbolic computations in high dimensions. The framework unifies invariant variational calculus with global differential invariants, offering robust tools for geometro-variational problems in both classical and relativistic settings.

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 (Möbius) groups and relate to some recent applications.

On globally invariant Euler--Lagrange equations for curves

TL;DR

This work develops a comprehensive global invariant calculus for curves, enabling Euler-Lagrange equations to be written purely in terms of differential invariants under Lie group actions. By constructing an invariant coframe and leveraging both cross-sections and moving frames, it derives an invariant EL operator via matrices and a relative invariant , applicable across Euclidean, Möbius, projective, and conformal geometries, including 4D Minkowski. The authors demonstrate both local and global invariant EL formulations for a variety of geometries, providing explicit formulas for Lagrangians such as , , , and others, and analyze singular extremals where the relative invariant degenerates (e.g., conformal geodesics). They also discuss computational challenges and propose a cross-section–based approach that complements moving frames, highlighting practical strategies for symbolic computations in high dimensions. The framework unifies invariant variational calculus with global differential invariants, offering robust tools for geometro-variational problems in both classical and relativistic settings.

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 (Möbius) groups and relate to some recent applications.
Paper Structure (37 sections, 6 theorems, 160 equations, 1 table)

This paper contains 37 sections, 6 theorems, 160 equations, 1 table.

Table of Contents

  1. Introduction
  2. Invariant Variational Problems
  3. Jets of curves
  4. Invariant differential forms
  5. Curves in the Euclidean plane
  6. The $\mathcal{C}$-spectral sequence
  7. The variational bicomplex
  8. The invariant variational complex
  9. Curves in the Euclidean plane
  10. Computing invariant Euler--Lagrange equations with moving frames
  11. Curves in the Euclidean plane
  12. Global invariants via cross-section
  13. Computations for classical geometries
  14. Möbius Invariant Variational Problems in 2D
  15. Generating differential invariants
  16. ...and 22 more sections

Key Result

Theorem 1

Let $G$ be an algebraic Lie pseudogroup of diffeomorphisms that acts transitively on $M$. The field of rational absolute differential invariants separates orbits in general position. Furthermore, this field is finitely generated in the sense that there exists finitely many rational differential inva

Theorems & Definitions (20)

  • Definition 1
  • Theorem 1: The global Lie--Tresse theorem kruglikov2016global
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Remark 2
  • Remark 3
  • ...and 10 more