Symmetries and currents in nonholonomic mechanics

Čech, Michal ; Musilová, Jana

Communications in Mathematics, Tome 22 (2014) no. 2, pp. 159-184.

Résumé

In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation generator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion. Moreover, the expressions for two other currents are obtained. Remarkably, the corresponding constraint symmetries are not symmetries of nonholonomic equations of motion. The physical interpretation of results is emphasized.

Détail

Classification : 49S05, 58E30
Keywords: nonholonomic mechanical systems; nonholonomic constraint submanifold; canonical distribution; reduced equations of motion; symmetries of nonholonomic systems; conservation laws; Chaplygin sleigh