Energy functional

Definition

Fix a Riemannian manifold. The energy functional is a map from the space of piecewise smooth curves on the manifold, to reals, defined as follows. The energy functional of a curve $\gamma :[0,1]\to \mathbb{R}$ is:

$\underset{0}{\overset{1}{\int }}{\left|\frac{d\gamma }{dt}\right|}^{2}dt$

The energy functional is thus parametrization-dependent, viz if we take an increasing function $f:[0,1]\to [0,1]$ the energy functional evaluated at $\gamma$ and at $f\circ \gamma$ may give different values.

For a given curve, the reparametrization for which the energy functional is minimized is when it is parametrized by arc-length.

Related notions

• Arc-length functional
• Generalized energy functional