On the existence of singularities in the geometrization of lagrangian dynamics

It is shown that the standard geometric picture of an important class of nonrelativistic Lagrangian motions has the origin of the generalized velocity space as a singular point. This occurs when the motion's generating force has a less than quadratic dependence on the generalized velocities. The importance cases of a gradient force-field and that of Rayleigh force-field are considered as exemples. The corresponding dynamical connections are constructed and present poles of order two one, respectively, at the origin of velocity space. This implies that well-behaved Lagrangian dinamics may originate ill-behaved gauge-fields in configuration space. (author)