A NONLINEAR CONNECTION FOR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS

A geometric setting for studying higher order ordinary differential equations (HODE) is obtained by choosing a nonlinear connection associated to the HODE. One such nonlinear connection was introduced in local coordinates by Miron and Atanasiu [23]. In this note we summarize results from [8], and show that this nonlinear connection has many of the useful properties of the canonical nonlinear connection associated to a system of second order differential equations. For example, (1) the nonlinear connection has simple coordinate free characterizations, (2) using the connection, key objects like the Jacobi endomorphism, dynamical co-variant derivatives and variation equations can be written using simple equations and (3) the connection can be used to geometrically express some of the invariants that appear in problems like equivalence problems.