Geodesics are exactly critical points of energy with fixed endpoints

If $\gamma$ is geodesic, [F1] gives $A=0$. Thus [F2] gives zero first variation for every smooth fixed-endpoint variation.

Conversely, assume every such first variation is zero and fix $t_0\in (a,b)$. Choose a coordinate chart $x:U\to x(U)$ at $\gamma (t_0)$. Some Euclidean ball $B_{3r}(x(\gamma (t_0)))$ lies in $x(U)$; after decreasing $h>0$, $[t_0-2h,t_0+2h]\subset (a,b)$ and $x(\gamma (t))\in B_r(x(\gamma (t_0)))$ throughout that interval. Applying [F3] in $\mathbb{R}$ and translating gives $\chi :\mathbb{R}\to [0,1]$ with $\chi =1$ on $[t_0-h,t_0+h]$ and support in $(t_0-2h,t_0+2h)$.

Put $W(t)=\chi (t)A(t)$. This is a smooth field along $\gamma$, supported in the chart interval and zero on neighborhoods of its two ends. Let $w(t)$ be its coordinate components there. The continuous function $\mathrm{\mid }w(t)\mathrm{\mid }$ has a maximum $C$ on the compact closed interval by [F3]. For $\mathrm{\mid }s\mathrm{\mid }<r\mathrm{/}(C+1)$ define $\alpha (s,t)=\{\begin{array}{ll}{x}^{-1}(x(\gamma (t))+sw(t)),& t\in (t_0-2h,t_0+2h),\\ \gamma (t),& t\mathrm{\notin }\mathrm{supp}\chi \mathrm{.}\end{array}$ The two formulas agree on neighborhoods of the gluing points. Moreover $\mathrm{\mid }sw(t)\mathrm{\mid }<r$, so the perturbed coordinate lies in $B_{2r}(x(\gamma (t_0)))\subset x(U)$. Hence $\alpha$ is a smooth fixed-endpoint variation with variation field $W$.

Applying the criticality assumption and [F2] to step 2.1 gives $0=-{\int }_a^b\chi (t)\mathrm{\mid }A(t){\mathrm{\mid }}_g^{2}dt\mathrm{.}$ The integrand is nonnegative and continuous, and at $t_0$ it equals $\mathrm{\mid }A(t_0){\mathrm{\mid }}_g^2$. If $A(t_0)\mathrm{\ne }0$, [F4] contradicts the displayed zero integral. Therefore $A(t_0)=0$. Since $t_0$ was arbitrary, $A=0$ on $(a,b)$, and smooth one-sided extension gives $A(a)=A(b)=0$ as well. Thus [F1] makes $\gamma$ a geodesic.

Steps 1.1 and 3.1 prove both implications. Constant curves have $A=0$. In dimension zero every curve is locally constant, so both sides hold without the positive-dimensional coordinate construction; dimension one is exactly the one-variable case above. An empty $M$ supplies no curve. The condition $a<b$ provides interior test points, while endpoint acceleration follows by smooth one-sided continuity. For each fixed $t_0$ only one chart, two radii, and one explicit bump are used; no simultaneous selection over all $t_0$ is made, so no choice axiom is needed.