First variation of geodesic energy
= First variation of geodesic energy
{title2=$dE_\gamma(V)=-\int\langle V,D_t\dot\gamma\rangle$}
For $E(\gamma)=\tfrac12\int_0^1|\dot\gamma|^2dt$ and <variation vector field> $V$, the first variation is $[\langle V,\dot\gamma\rangle]_0^1-\int_0^1\langle V,D_t\dot\gamma\rangle dt$. For fixed endpoints the boundary term vanishes, so the <critical points> of the energy are exactly affinely parametrized <geodesics>.