Geodesic equals energy critical path: Difference between revisions
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Let <math>M</math> be a [[complete Riemannian manifold]]. Then the piecewise smooth [[geodesic in a Riemannian manifold|geodesic]]s in <math>M</math> are precisely the same as the [[critical path of a functional|critical paths]] of the [[energy functional]]. | Let <math>M</math> be a [[complete Riemannian manifold]]. Then the piecewise smooth [[geodesic in a Riemannian manifold|geodesic]]s in <math>M</math> are precisely the same as the [[critical path of a functional|critical paths]] of the [[energy functional]]. | ||
==Proof== | |||
===Geodesic implies critical path of energy functional=== | |||
This is clear -- any geodesic is locally length-minimizing, and hence, locally energy-minimizing, and hence, it must be a critical path of the energy functional. | |||
===Critical path of energy functional implies geodesic=== | |||
The proof of this follows by cleverly choosing a variation, and then applying the [[first variation formula]] to conclude, from the vanishing of the derivative, that the acceleration vector must vanish at each point. | |||
Latest revision as of 19:41, 18 May 2008
Statement
Let be a complete Riemannian manifold. Then the piecewise smooth geodesics in are precisely the same as the critical paths of the energy functional.
Proof
Geodesic implies critical path of energy functional
This is clear -- any geodesic is locally length-minimizing, and hence, locally energy-minimizing, and hence, it must be a critical path of the energy functional.
Critical path of energy functional implies geodesic
The proof of this follows by cleverly choosing a variation, and then applying the first variation formula to conclude, from the vanishing of the derivative, that the acceleration vector must vanish at each point.