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?

  • 1
    $\begingroup$ Sounds interesting, what's the functional? $\endgroup$
    – MBN
    Mar 22, 2011 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, 2011 at 18:55

1 Answer 1


Yes there is, the curve is a a hypocycloid.

See for instance:





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