Geodesic equals energy critical path
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.