Geometry of a pair of second-order ODEs and Euclidean spaces

Abstract

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler-Lagrange equations of the arclength action. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a pair of second-order ordinary differential equations be reparameterized so as to give, locally, the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second-order ODEs revealing the existence of 24 invariant functions.