The Euler-Lagrange PDE and Finsler metrizability

TL;DR

This paper explores conditions under which second-order differential equations can be derived from Finsler metrics, providing a reduction of the Euler-Lagrange PDE to a first-order system and offering criteria for Finsler metrizability.

Contribution

It introduces a reduction of the Euler-Lagrange PDE to a first-order system and establishes necessary and sufficient conditions for Finsler metrizability based on holonomy algebra.

Findings

Reduced the Euler-Lagrange PDE to a first-order system

Provided criteria for local existence of Finsler metrics

Extended results to Landsberg metrizability problem

Abstract

In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove similar results. This reduction is a significant step in solving the problem whether or not there exists a non-Berwald Landsberg space.