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arXiv · 1109.3640

Homogeneity and projective equivalence of differential equation fields

M. Crampin·D. J. Saunders
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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.

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M. Crampin, D. J. Saunders. 2011-09-16. Homogeneity and projective equivalence of differential equation fields. https://arxiv.org/abs/1109.3640

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