 Solution

A geodesic with affine parameter is a curve whose velocity is parallel along itself, ${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$. Its energy on a fixed parameter interval is $E(\gamma )=\frac{1}{2}\int ⟨\dot{\gamma },\dot{\gamma }⟩dt$. For a smooth fixed-endpoint variation with variation field $V$, metric compatibility and the torsion-free connection giveThe endpoint term vanishes. Thus a geodesic is a critical point; conversely, vanishing for every interior variation field implies ${\nabla }_t\dot{\gamma }=0$. Affinely parametrized geodesics are precisely the fixed-endpoint energy critical points.