This recent question about holes dug through the Earth led me to wonder: if I wanted to dig out a tube from the north pole to the equator and build a water slide in it, which shape would be the fastest?
We're assuming a frictionless tube, of course. Let's ignore centrifugal forces. Coriolis forces do no work, and so shouldn't matter. Also, let's assume the Earth is a sphere with uniform density.
I tried to solve this problem by writing down an integral in polar coordinates for the time, then applying the Euler-Lagrange equations. However, I didn't make any progress on the resulting differential equation. Is there an analytical expression for the curve?