  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.
