= Moving-endpoint second variation of Riemannian arc length
{title2=$L''(0)=[\langle A,T\rangle]_a^b+I(J^\perp,J^\perp)$}
For a variation of a unit-speed <geodesic>, let $T$ be its tangent, $J$ its <variation vector field> and $A=\nabla_sJ$ its variation acceleration. The <second variation of Riemannian arc length> has the displayed endpoint term in addition to the <Riemannian index form> of $J^\perp=J-\langle J,T\rangle T$. Differentiating the speed twice gives $|D_tJ|^2-\langle D_tJ,T\rangle^2+\langle D_sD_tJ,T\rangle$. Commuting derivatives and integrating $D_tA$ by parts gives the formula. The term cannot be discarded unless the endpoints are fixed or an appropriate endpoint condition makes it zero.
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