Nonholonomic Mechanical Systems and Kaluza-Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper we show that the Kaluza– Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projection on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. Moreover, such a geodesic projects exactly on a solution of the Lorentz force equations of the spacetime. Finally, by using nonholonomic geometry we present the induced Einstein equations on the spacetime, by the Einstein equation on a Kaluza–Klein space with stress–energy tensor. AMS Subject Classification: 70F25, 83E15.