Moving-endpoint second variation of Riemannian arc length ($L^{\prime \prime }(0)=[⟨A,T⟩{]}_a^b+I(J^{\perp },J^{\perp })$)

For a variation of a unit-speed geodesic, let $T$ be its tangent, $J$ its variation vector field and $A={\nabla }_{s}J$ its variation acceleration. The second variation of Riemannian arc length has the displayed endpoint term in addition to the Riemannian index form of $J^{\perp }=J-⟨J,T⟩T$. Differentiating the speed twice gives $|D_{t}J{|}^2-⟨D_{t}J,T{⟩}^2+⟨D_s{D}_{t}J,T⟩$. Commuting derivatives and integrating $D_{t}A$ by parts gives the formula. The term cannot be discarded unless the endpoints are fixed or an appropriate endpoint condition makes it zero.