Arc-length functional

Definition

Fix a Riemannian manifold $M$. The arc-length functional is a map from the space of piecewise smooth curves in the manifold, to real numbers, defined as follows. The arc-length of a curve $\gamma :[0,1]\to M$ is:

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

Interestingly, the arc-length is independent of the parametrization of the curve. In other words , if $f:[0,1]\to [0,1]$ is an increasing function, then $\gamma \circ f$ has the same arc-length as $\gamma$.

Related notions

• Energy functional
• Generalized energy functional