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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?

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Sounds like a hard problem! –  Ted Bunn Mar 22 '11 at 18:31
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Sounds interesting, what's the functional? –  MBN Mar 22 '11 at 18:45
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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 –  Qmechanic Mar 22 '11 at 18:55
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1 Answer

up vote 6 down vote accepted

Yes there is, the curve is a a hypocycloid.

See for instance:

http://en.wikipedia.org/wiki/Hypocycloid

http://demonstrations.wolfram.com/SphereWithTunnelBrachistochrone/

http://www.physics.unlv.edu/~maxham/gravitytrain.pdf

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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
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(The first of my three Wolfram links is just for the ordinary Brachistochrone problem + friction.) –  Qmechanic Mar 25 '11 at 21:30
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