Geodesic equals energy critical path: Difference between revisions

Revision as of 10:49, 5 August 2007

Statement

Let $M$ be a complete Riemannian manifold. Then the piecewise smooth geodesics in $M$ 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.

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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]].
+==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.