Geometric control theory provides the calculus of variations new perspectives that both unify its classic theory and outline new horizons toward which its theory extends. These perspectives grow from the theoretical foundations anchored in two important theorems not available to the classic theory of the calculus of variations.

The more immediate of these two theorems is the “maximum principle” of L. S. Pontryagin and his co-workers, obtained in the late 1950s. The maximum principle, a far-reaching generalization of Weierstrass's necessary conditions for strong minima, provides geometric conditions for a (strong) minimum of an integral criterion, called the “cost,” over the trajectories of a differential control system. These conditions are based on the topological fact that an optimal solution must terminate on the boundary of the extended reachable set formed by the competing curves and their integral costs.

An important novelty of Pontryagin's approach to problems of optimal control consists of liberating the variations along the optimal curves from the constricting condition that they must terminate at the given boundary data. Instead, he considers variations that are infinitesimally near the terminal point and that generate a convex cone of directions locally tangent to the reachable set at the terminal point defined by the optimal trajectory. As a consequence of optimality, the direction of decreasing cost cannot be contained in the interior of this cone. This observation leads to the “separation theorem,” which can be seen as a generalization of the classic Legendre transform in the calculus of variations, which ultimately produces the appropriate Hamiltonian function.