In this chapter, we explore a formalism of classical mechanics that is in some sense dual to the Hamiltonian one introduced in the last chapter by focussing on velocity instead of momentum. This formalism initially preceded the Hamiltonian one and gave rise to the first description of motions via an extremal action principle of the calculus of variations. In particular, we introduce the Lagrangian and the Hamiltonian action functional, and show how critical points thereof give rise to motions in physical systems. We outline some advantages and disadvantages of the two approaches.