On geometrization of motion of some nonholonomic systems

In this note we consider mechanical systems with smooth linear nonholonomic constraints which do not depend on time. For a special case of Chaplygin systems, when the motion of the system can be described as a closed system of differential equations in local coordinates of the reduced space of the same dimension as the dimension of the constraint distribution, we define linear connections for which the equations of motion take the form of equations of geodesic lines. In the case of inertial motion, or motion under influence of potential forces, we give explicit expressions for coefficients of linear connections, in a form much simpler then those given for a general case in [4]. Let us consider mechanical system M with local coordinates q (i = 1, . . . , n). It is well known that the configuration space Vn of M is the Riemannian space with the metric defined by the expression