
The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces
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 secondorder differential equations: the
EulerLagrange equations of the metric. 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 given pair of second order ordinary
differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and
$d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by
$t\rightarrow T(y,t)$ 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 ordinary
differential equations of the above form revealing the existence
of 24 invariant functions.
Category:34A26 