The Geometry of Mixed First and Second - Order D

A geometrical framework is presented for a class of dynamical systems, which are modelled by a mixed system of rst and second-order ordinary diierential equations. The starting point for the model is a bundle : E ! M, where both E and M are bred over IR. We show that there are two connections associated to systems of this kind: the rst one comes from thèconstraint equations' (the rst-order equations); the second one is related to the full system on the constraint manifold. Among other things, we discuss the concepts of symmetry and adjoint symmetry for such systems and identify for that purpose an appropriate notion of`dynamical covariant derivative' and`Jacobi endomorphism'. The curvature tensors of the two connections play an important role in deening these concepts. The present approach in particular is relevant for modelling certain mechanical systems with non-holonomic constraints.