Energy functional

From Diffgeom
Jump to: navigation, search

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