On the dinamics of nonholonomic systems

Abstract

In the development of nonholonomic mechanics one can observe recurring confusion over the very equations of motion as well as the deeper questions associated with the geometry and analysis of these equations. First of all, as far as the equations of motion themselves are concerned, the confusion mainly centered on whether or not the equations could be derived from a variational principle in the usual sense. Attempting to dissipate this confusion, in the present paper we deduce a new form of equations of motion which are suitable for both nonholonomic systems with either linear or nonlinear constraints and holonomic systems (A-model). These equations are deduced from the principle of stationary action (or Hamiltonian principle) with nonzero transpositional relations. We show that the well-known equations of motion for nonholonomic and holonomic systems can be deduced from the A-model. For the systems which we call the generalized Vorones-Chaplygin systems we deduce the equations of motion which coincide with the Vorones and Chaplygin equations for the case in which the constraints are linear with respect to the velocity. An additional result is that the transpositional relations are different from zero only for those generalized coordinates whose variations (in accordance with the equations of nonholonomic constraints) are dependent. For the remaining coordinates, the transpositional relations may be zero.