Some geometric aspects of variational calculus in constrained systems

We give a geometric description of variational principles in constrained mechanics. For the general case of nonholonomic constraints, a unified variational approach to both vakonomic and nonholonomic frameworks is given, and the corresponding equations of motion are recovered. Special attention is paid to the existence of infinitesimal variations in both cases, and it is proved that these variations coincide when the constraints are integrable. As examples, we give geometric formulations of the equations of motion for the case of optimal control and for vakonomic and nonholonomic mechanics with constraints linear in the velocities.