One can compute the second variation for both energy and length. There is a factor different in ther intergration term. $h(s,t)$ is the (proper) variation, $\forall t, h(0,t)=c(t)$, and let $Y=D_{(0,t)}h(\partial_s)$. In the second variation for length and engergy, the term mostly coincide. Except for a term in the intergral, while the second variation for energy is $Y$, but length is $Y^\perp$ (components perpendicular to $c'$).
But on the other hand, if it is unit geodesic, also wlog assume b-a=0.5 then $L=\sqrt{E}$. So $L'=\frac{E'}{\sqrt{E}}$, $L''=\frac{E''\sqrt{E}-\frac{E'^2}{2\sqrt{E}}}{E}$. Because E'=0, and $E$ is fixed, $L''$ is kinda porpotional to $E''$. However, it is not. Why is it like that?