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Symmetries and currents in nonholonomic mechanics. (English) Zbl 1308.49045
Summary: 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 a 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 the results is emphasized.

MSC:
49S05 Variational principles of physics
49Q99 Manifolds and measure-geometric topics
70F25 Nonholonomic systems related to the dynamics of a system of particles
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
58E30 Variational principles in infinite-dimensional spaces
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