The calculus of variations in the large studies stationary values and the corresponding extremals of a functional using the global topology of its space of admissible functions or curves (Morse 1934). It concerns existence and multiplicity as well as the local behavior of an individual extremal.
For example, the stationary paths of the energy functional on paths joining two fixed points in a Riemannian manifold are constant-speed geodesics. In a finite-dimensional approximation by piecewise geodesics, Morse theory relates changes in the topology of the sublevel sets of energy to its critical points. The Morse index is the largest dimension of a subspace on which the second variation is negative definite. This approach ties the existence of geodesics to the topology of the path space (Milnor 1963, Part III).
