Energy functional on paths

Summary

The energy of a path in a Riemannian manifold is half the integral of the squared speed. Among paths with fixed ends, its critical points are exactly the geodesics with constant speed. In variational work it is preferred to the length. It is not invariant under reparametrization, so its critical points are better behaved and its Hessian is easier to compute. It is the main example of a function to which Morse theory on infinite-dimensional manifolds applies.