= 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 $E(\gamma)=\frac12\int\langle\dot\gamma,\dot\gamma\rangle\,dt$. For a smooth fixed-endpoint variation with variation field $V$, metric compatibility and the torsion-free connection give
$$
\delta E=\int\langle\nabla_tV,\dot\gamma\rangle\,dt
=[\langle V,\dot\gamma\rangle]_{t_0}^{t_1}
-\int\langle V,\nabla_t\dot\gamma\rangle\,dt.
$$
The endpoint term vanishes. Thus a <geodesic> is a critical point; conversely, vanishing for every interior variation field implies $\nabla_t\dot\gamma=0$. \b[Affinely parametrized <geodesics> are precisely the fixed-endpoint energy critical points.]
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