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?

| cite | improve this question | | | | |
  • 1
    $\begingroup$ Sounds interesting, what's the functional? $\endgroup$ – MBN Mar 22 '11 at 18:45
  • 2
    $\begingroup$ It seems like this article should be relevant: H. L. Stalford and F. E. Garrett, "Classical differential geometry solution of the brachistochrone tunnel problem", Journal of Optimization Theory and Applications, Volume 80, Number 2, 227-260, springerlink.com/content/f21724177qxptn56 $\endgroup$ – Qmechanic Mar 22 '11 at 18:55

Yes there is, the curve is a a hypocycloid.

See for instance:




| cite | improve this answer | | | | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.