How to minimize the length of a curve on $S^2$

Question

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi (d\theta/dt)^2}dt$$ I want to minimize the distance using calculus. How do we do that? What do we differentiate with respect to and equate to $0$? I am not able to start.

Answer 1

You should consider a vectorfield $v$ defined on the points of $\gamma$, tangent to the sphere, orthogonal to the curve and zero on the two extremes of the curve. Then consider the variation of your curve $\gamma$: $$ \gamma_h (t) = \gamma(t) + h v $$ Then write your functional $\ell_{\gamma_h}$ and compute the derivative with respect to $h$ for $h=0$. Then you let this derivative be equal to $0$ for all $v$.