Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
Sounds interesting, what's the functional? – MBN Mar 22 '11 at 18:45
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, – Qmechanic Mar 22 '11 at 18:55
up vote 6 down vote accepted

Yes there is, the curve is a a hypocycloid.

See for instance:

share|cite|improve this answer
Wow! Surprising result - thanks for the links. I'll have to read over all the details later today. – Mark Eichenlaub Mar 22 '11 at 19:48
Very nice maths. Otherwise, concerning practical applications, it's handy! ;-) – Luboš Motl Mar 22 '11 at 20:57
Very good answer. – Andrew Mar 22 '11 at 21:29
(The first of my three Wolfram links is just for the ordinary Brachistochrone problem + friction.) – Qmechanic Mar 25 '11 at 21:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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