Abstract
A general notion of connections over a vector bundle map is considered, and applied to the study of mechanical systems with linear nonholonomic constraints and a Lagrangian of kinetic energy type. In particular, it is shown that the description of the dynamics of such a system in terms of the geodesics of an appropriate connection can be easily recovered within the framework of connections over a vector bundle map. Also the reduction theory of these systems in the presence of symmetry is discussed from this perspective.