Let $(M,g)$ be a pseudo-riemannian manifold and $p,q\in M$. Suppose $\alpha:[a,b]\to M$ a smooth curve on $M$ such that $\alpha(a)=p$ and $\alpha(b)=q$. If we consider: $$L[H(s,\cdot)]=\int_a^b \sqrt{g_{H(s,t)}(\partial_t H,\partial_tH)}$$ as a function of $s$, where $H:(-\varepsilon,\varepsilon)\times (a-\varepsilon,b+\varepsilon)\longrightarrow \mathbb{R}$ is an smooth map which satisfies:
(i) $H(0,\cdot)=\alpha$ in $[a,b]$
(ii) $H(s,a)=p$ and $H(s,b)=q$.
My question is:
If I prove that there exists an unique $\alpha_0$ (exept reparametrization) such that: $$\frac{d}{ds}\Big(L[H(s,\cdot)]\Big)_{s=0}=0$$ How can I conclude that does not exists any $\beta$ curve in $M$ with $\beta(a)=p$ and $\beta(b)=q$ s.t. $long(\alpha)=long(\beta)$?
I can't write exactly why this happend. Many thanks!