Time-dependent Lagrangian systems: A geometric approach to the theory of systems with constraints

A geometric approach to the theory of time-dependent regular Lagrangian systems with constraints is presented using the framework of the exact contact manifold (TQ×R,ΘL). The main subject of the article concerns the properties of the time-dependent nonholonomic constraints and the geometric approach to the method of the Lagrange multipliers. It is shown that every constraint determines a contact one-form, a vertical vector field, and a nonvertical vector field. The explicit form of the vector field representing the constrained dynamics is obtained and, finally, the properties of all these one-forms and vector fields are discussed.