Geodesic equals energy critical path

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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.


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.