Generalized energy functional: Difference between revisions

Revision as of 08:41, 5 August 2007

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 $γ$ is:

$\int_{0}^{1} | \frac{d γ}{d t} | d t$

Particular cases

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

@@ Line 4: / Line 4: @@
 <math>\int_0^1 \left|\frac{d\gamma}{dt}\right| dt</math>
+==Particular cases==
+For <math>p=1</math>, we get the [[arc-length functional]] and for <math>p=2</math>, we get the [[energy functional]].