For simplicity without loss of generalization, consider a free particle.
When using the Principle of Least Action, I imagine all variations of the true path between $t_1, t_2$ regardless of whether they're possible or not. But whereas some possible paths can be realized by varying the initial and corresponding final velocities, others are going to require an additional virtual potential $\delta V(q)$ to give the particle the corresponding force required. Yet the potential is always assumed to be zero in the Lagrangian when evaluating the action in this case.