Nonlocal interpretation of $\lambda$-variational symmetry-reduction method

TL;DR

This paper provides a geometric interpretation of the $\lambda$-variational symmetry reduction method, framing it as a nonlocal symmetry approach that enhances reduction of variational ODEs lacking local symmetries.

Contribution

It offers a geometric perspective on the $\lambda$-variational symmetry method, clarifying its role as a nonlocal symmetry reduction technique.

Findings

The method is better understood as a nonlocal symmetry reduction.

It is particularly effective for variational ODEs with few local symmetries.

Provides insights into partial reduction capabilities.

Abstract

In this paper we give a geometric interpretation of a reduction method based on the so called $\lambda$-variational symmetry (C. Muriel, J.L. Romero and P. Olver 2006 \emph{Variational $C^{\infty}$-symmetries and Euler-Lagrange equations} J. Differential equations \textbf{222} 164-184). In general this allows only a partial reduction but it is particularly suitable for the reduction of variational ODEs with a lack of computable local symmetries. We show that this method is better understood as a nonlocal symmetry-reduction.