Energy functional: Difference between revisions
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==Definition== | ==Definition== | ||
Fix a Riemannian manifold. The energy functional is a map from the space of piecewise smooth | Fix a Riemannian manifold. The energy functional is a map from the space of [[piecewise smooth path]]s on the manifold, to reals, defined as follows. The energy functional of a curve <math>\omega:[0,1] \to M</math> is: | ||
<math>\int_0^1 \left|\frac{d\ | <math>\int_0^1 \left|\frac{d\omega}{dt}\right|^2 dt</math> | ||
The energy functional is thus parametrization-dependent, viz if we take an increasing function <math>f:[0,1] \to [0,1]</math> the energy functional evaluated at <math>\ | The energy functional is thus parametrization-dependent, viz if we take an increasing function <math>f:[0,1] \to [0,1]</math> the energy functional evaluated at <math>\omega</math> and at <math>\omega \circ f</math> 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. | For a given curve, the reparametrization for which the energy functional is minimized is when it is parametrized by arc-length. | ||
Revision as of 08:50, 5 August 2007
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 is:
![{\displaystyle \omega :[0,1]\to M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73b9fbd934563330f8782e88c6d398cabd83402a)
The energy functional is thus parametrization-dependent, viz if we take an increasing function the energy functional evaluated at and at may give different values.
![{\displaystyle f:[0,1]\to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f64956bbc19d5bbe642fd7bf4f1aa62487006f4d)
For a given curve, the reparametrization for which the energy functional is minimized is when it is parametrized by arc-length.