The Weitzenböck connection and time reparameterization in nonholonomic mechanics

Abstract

We show that the torsion of the Weitzenböck connection is responsible for the fictitious pseudogyroscopic force experienced by a general mechanical system in a noncoordinate moving frame. In particular, we show that for the class of mechanical systems subjected to nonintegrable constraints known as non-abelian nonholonomic Chaplygin systems, the constraint reaction force directly depends on this torsion. For these Chaplygin systems, we show how this torsional force can in some cases be removed by an appropriate choice of frame depending on a multiplier f(q), linking these results to the process of Chaplygin Hamiltonization through time reparameterization. Lastly, we show that the cyclic symmetries of f in some cases lead to the existence of momentum conservation laws for the original nonholonomic system and illustrate the results through several examples.