Projective geometry of systems of second-order differential equations

Abstract

It is proved that every projective connection on an -dimensional manifold is locally defined by a system of second-order ordinary differential equations resolved with respect to the second derivatives and with right-hand sides cubic in the first derivatives, and that every differential system defines a projective connection on . The notion of equivalent differential systems is introduced and necessary and sufficient conditions are found for a system to be reducible by a change of variables to a system whose integral curves are straight lines. It is proved that the symmetry group of a differential system is a group of projective transformations in -dimensional space with the associated projective connection and has dimension . Necessary and sufficient conditions are found for a system to admit the maximal symmetry group; basis vector fields and structure equations of the maximal symmetry Lie algebra are produced. As an application a classification is given of the systems of two second-order differential equations admitting three-dimensional soluble symmetry groups.