Homogeneity and projective equivalence of differential equation fields

TL;DR

This paper generalizes the concepts of homogeneity and projective equivalence for higher-order differential equations, linking them to Euler-Lagrange fields and geodesic paths, thus extending classical geometric ideas.

Contribution

It introduces new definitions for homogeneity and projective equivalence in higher-order ODEs and connects these to Euler-Lagrange fields and geodesic systems.

Findings

Euler-Lagrange fields of regular parametric Lagrangians form a projective class

Homogeneous differential equations determine systems of paths

Geodesics are invariant under reparametrization within the class

Abstract

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are regular (in a natural sense that we define) form a projective equivalence class of homogeneous systems. We show further that the geodesics, or base integral curves, of projectively equivalent homogeneous differential equation fields are the same apart from orientation-preserving reparametrization; that is, homogeneous differential equation fields determine systems of paths.