A differential geometric setting for mixed first- and second-order ordinary differential equations

A geometrical framework is presented for modelling general systems of mixed first- and second-order ordinary differential equations. In contrast to our earlier work on nonholonomic systems, the first-order equations are not regarded here as a priori given constraints. Two nonlinear (parametrized) connections appear in the present framework in a symmetrical way and they induce a third connection via a suitable fibred product. The space where solution curves of the given differential equations live, singles out a specific projection p among the many fibrations in the general picture. A large part of the paper is about the development of intrinsic tools-tenser fields and derivations-for an adapted calculus along p. A major issue concerns the extent to which the usual construction of a linear connection associated with second-order equations fails to work in the presence of coupled first-order equations. An application of the ensuing calculus is presented.