The inverse problem for invariant Lagrangians on a Lie group

TL;DR

This paper investigates the inverse problem of finding invariant Lagrangians for systems of invariant second-order differential equations on Lie groups, using reduction techniques and Helmholtz conditions.

Contribution

It introduces a reduction method from the tangent bundle to the Lie algebra to analyze the inverse problem for invariant Lagrangians.

Findings

Reduction of the problem to the Lie algebra simplifies analysis.

Provides conditions for the existence of invariant Lagrangians.

Includes illustrative examples demonstrating the approach.

Abstract

We discuss the problem of the existence of a regular invariant Lagrangian for a given system of invariant second-order differential equations on a Lie group $G$, using approaches based on the Helmholtz conditions. Although we deal with the problem directly on $TG$, our main result relies on a reduction of the system on $TG$ to a system on the Lie algebra of $G$. We conclude with some illustrative examples.