I recently noticed that the classic brainteaser Carter's killer rabbit looks like a problem that can be solved with calculus of variations. I think it would be easiest in polar coordinates. Here is my modification:
Assume a rabbit starts in the center of a pond (circle with radius $R$) and an agent in position $(R,\frac{\pi}{2})$. The rabbit will start its movement towards $(R,\frac{3\pi}{2})$. The agent is $k$ times faster than the rabbit. Find the optimal path for the rabbit so it escapes from the agent as fast as possible. Let the position and speed be:
\begin{align*} x_{r}(t) &= (r(t), \theta_{r}(t)) \\ x_{a}(t) &= (R, \theta_{a}(t)) \\ \dot{x}_r &= (\dot{r}(t), \dot{\theta}_r(t)) \\ \dot{x}_a &= (0, k|\dot{x}_r|) \end{align*}
Now, I would like to solve this problem using calculus of variations. It looks to me similar to a Brachistochrone problem where the gravity, i.e. the agent, is moving in a circular path.
For backround purpose, here is the original problem:
Suppose, the day after attacking President Carter, the rabbit finds itself alone in the middle of the pond, which is perfectly circular. Suppose there is a single Secret Service agent on the edge of the pond, armed with a small net to ensnare the swimming rabbit as it approaches the edge. This net is effective only if the rabbit is still in the water. If the rabbit reaches any point on the edge before the agent does, it can hop away to freedom; if the agent gets there first, the rabbit will be captured.
If the agent runs four times as fast as the rabbit swims, can the rabbit escape? If so, how?
A similar question: Pursuit Curve. Dog Chases Rabbit. Calculus 4.