Why are normal sub-Riemannian extremals locally minimizing?

It is well-known that normal extremals in sub-Riemannian geometry are curves that locally minimize the length functional (equivalently, the energy functional). Most proofs of this fact do not make, however, an explicit use of relations between local optimality and the geometry of the problem. In this paper, we provide a new proof of that classical result, which gives insight into direct geometric reasons for local optimality. Also the relation of the regularity of normal extremals with their optimality becomes apparent in our approach.