I'm studying calculus of variations. I want to find the shortest path between two points on sphere. Let me talk specific example. On the sphere, choose $p = (0,1,0)$ and $q = (0,0,-1)$. Let $\mathbf{x}(u,v) = (\cos u \sin v, \sin u \sin v, \cos v)$ on $0 < u < 2\pi, 0<v<\pi$. Then $\mathbf{x}$ is a parametrization of the sphere containing $p$ and $q$. Let $\alpha : [\pi/2, \pi] \rightarrow (0,2\pi)\times (0,\pi)$ be a curve by $\alpha(t) = (t, v(t))$ and $v(\pi/2) = v(\pi) = \pi/2$. This curve connects $\mathbf{x}^{-1}(p)$ and $\mathbf{x}^{-1}(q)$ Then $\gamma = \mathbf{x} \circ \alpha$ is a curve on sphere connecting $p$ and $q$. And the arc length of $\gamma$ is \begin{equation} l = \int_{\pi/2}^\pi |\gamma'(t)| dt = \int_{\pi/2}^\pi \sqrt{E(t,v(t))+G(t,v(t))v'(t)^2}dt \text{ where $E$ and $G$ are the coefficients of first fundamental form.} \end{equation} Since $E(t,v(t)) = \sin^2v(t)$ and $G(t,v(t)) = 1$, \begin{equation} l = \int_{\pi/2}^\pi \sqrt{\sin^2 v(t)+v'(t)^2}dt \end{equation} Then let $f(v,v') = \sqrt{\sin^2v + (v')^2}$. Since $f$ does not depend on $t$, we can use the Beltrami equation to find minimizer of $l$, i.e, if the minimizer of $l$ exists, \begin{equation} f - v'\frac{\partial f}{\partial v'} = c, \text{ c is constant.} \end{equation} Since $v(0) = \pi/2$, let $v(t) = \pi/2$ for all $t$. Then this is a solution of the Beltrami equation.
But in the calculus of variations, satisfying this equation is just a necessary condition for minimizer. Additionally, Euler-Lagrange equation states that if $f(t,v,v')$ is convex for all $t$ and there is a solution of equation, then that solution is actually minimizer of that. So I tried to show $f(x,y)$ is convex, but it fails when I checked second-order partial derivative condition. So, I wonder how can I confirm, that is actual minimizer of the path connecting the given points.
And also I want to find easy example of finding shortest path in regular surface, which can be proved with using the calculus of variations. It means that given $f$ has convexity and Beltrami(or E-R equation) has explicit solution. For example, just plane surface. I hope it's not a surface like cylinder or cone. Also I know othe methods of finding shortest path on regular surface using geodesic, but I want to use the calculus of variation method. Thank you.