You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take?

Formally, fix spherical coordinates $(\theta, \psi)$ on $S^2$ with $\theta$ representing the latitute and $\psi$ the longitude. We suppose wlog we start at $p := (\theta_0, 0)$ for some angle $\theta_0 \geq 0$.

Denote by $N$ the North Pole, and fix $r > 0$ such that $r < d(N, p)$, where $d$ denotes the Riemannian distance. We consider the variational problem 

$$\inf \{ \psi(\gamma(1)) \, | \, \gamma \in C^1 ([0, 1], S^2) \,  | \, \gamma(0) = p, L(\gamma) = r \},$$

where $L(\gamma) := \int_0^1 |\gamma’(t)| \, dt$ denotes the arc length of $\gamma$.

**Question:** What is the optimal path $\gamma$, and what is the optimal value of the ending time zone $\psi(\gamma(1))$?