On the consistency of the Lagrange multiplier method in classical mechanics

Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so, one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion and can only depend on the generalized coordinates and velocities, as well as time. In this paper, we establish conditions the Lagrangian and the constraints should obey in order to guarantee that the constraint forces can be obtained consistently.