Energy functional: Difference between revisions
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==Definition== | ==Definition== | ||
The energy functional is a map from the space of piecewise smooth curves, to reals, defined as follows. The energy functional of a curve <math>\gamma:[0,1] \to \R</math> is: | 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 <math>\gamma:[0,1] \to \R</math> is: | ||
<math>\int_0^1 \left|\frac{d\gamma}{dt}\right|^2 dt</math> | <math>\int_0^1 \left|\frac{d\gamma}{dt}\right|^2 dt</math> | ||
Revision as of 08:39, 5 August 2007
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 is:
![{\displaystyle \gamma :[0,1]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eaf1c0ebc6e628b53498811c91b9df46e394d07)
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.