Freedom of h(2)-variationality and metrizability of sprays

TL;DR

This paper explores the conditions under which second order differential equations called sprays can be derived from variational principles, focusing on homogeneous Lagrangians of degree two and their geometric characterizations.

Contribution

It introduces a geometric method using holonomy distributions to determine the degree of h(2)-variationality of sprays, including isotropic sprays as an example.

Findings

Holonomy distribution characterizes h(2)-variationality.

The method applies to isotropic sprays.

Provides a geometric criterion for metrizability.

Abstract

In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the directional argument. Searching for geometric objects characterizing the degree of freedom of h(2)-variationality of a spray, we show that the holonomy distribution generated by the tangent direction to the parallel translations can be used to calculate it. As a working example, the class of isotropic sprays is considered.