Connections and Time Reperametrizations in Nonholonomic Mechanics

Abstract

We consider nonholonomic system (M,L,D) on configuration space M given with Lagrangian Land nonintegrable distribution D defined by linear nonholonomic constraints. The equations of motion are obtained from the Lagrange-d’Alembert principle. In classical works of Synge [14], Vranceanu [18], Shouten [13], Wagner [15,16] the problem of motion of nonholonomic systems from the geometric point of view is considered. The equations can be rewritten in terms of suitable vector bundle connection∇P over configuration space M: ∇P ̇q ̇q=−gradDV. In the case when the potential V vanishes, the solutions becomes the geodesic lines of ∇P. We recall on the extensions of the vector-bundle connection to the linear connection on TM considered in [3,17] and [12], as well as on so called partial connection (see [7]). We compare various approaches in geometrical formulation of nonolonomic systems by using affine connections, including the Chaplygin reduction performed by Bakša [1]. Although mentioned objects are very well studied, some natural relationships between them are pointed out. In addition, we consider the Newton type equations on a Riemannian manifold (M,g) and look for a conformal metric g∗=f2g such that solutions of the Newton equations, after a time reparametrization, become the geodesic lines of g. This is a generalization of the Chaplygin multiplier method for Hamiltonization of G-Chaplygin systems [4,5]. Also, we obtain variants of the Maupertuis principle in nonholonomic mechanics as they are given in [1,11].