Generalized energy functional

Definition

Fix a Riemannian manifold. The generalized energy functional is a map from the space of piecewise smooth curves in the manifold, to reals, defined as follows: the generalized energy functional evaluated at a curve $\gamma$ is:

$\underset{0}{\overset{1}{\int }}{\left|\frac{d\gamma }{dt}\right|}^{p}dt$

Particular cases

For $p=1$, we get the arc-length functional and for $p=2$, we get the energy functional.