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:

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

Particular cases

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