摘要：
In this thesis, we investigate the local properties of second-order ODE systems [special characters omitted] subject to point-transformations [special characters omitted] By defining the notion of a path geometry on a manifold as a system of unparameterized curves, corresponding to solutions of such an ODE system, one can apply É. Cartan's method of equivalence, which yields a complete set of local differential invariants. The flat model for path geometry consists of lines in projective space, and on any path geometry there is induced a structure generalizing the structure of projective space as a homogeneous space of the special linear group. With this structure, one can identify natural classes of ODE systems, such as those describing geodesics of an affine connection. Complementing this class of geodesic equations are a new class, called torsion-free path geometries, with remarkable geometric properties, including in the generic case an abundance of invariant first integrals. These are studied in detail in the case of pairs of ordinary differential equations, where one finds a twistor correspondence with half-flat, split-signature conformal structures. We additionally investigate, in the context of differential invariants, the special role played by ODE systems of Euler-Lagrange type; and we discuss the relationship between second-order ODE systems subject to point transformations and a certain natural class of PDE systems subject to gauge transformations.
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