Generalized energy functional

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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.