Fix a Riemannian manifold. The energy functional is a map from the space of piecewise smooth paths on the manifold, to reals, defined as follows. The energy functional of a curve is:
The energy functional is thus parametrization-dependent, viz if we take an increasing function the energy functional evaluated at and at may give different values.
For a given curve, the reparametrization for which the energy functional is minimized is when it is parametrized by arc-length.