Energy functional

Definition

Fix a Riemannian manifold. The energy functional is a map from the space of piecewise smooth paths on the manifold, to reals, defined as follows. The energy functional of a curve $\omega :[0,1]\to M$ is:

$\int _{0}^{1}\left|{\frac {d\omega }{dt}}\right|^{2}dt$

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 $\omega$ and at $\omega \circ f$ 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