Sunday, November 4, 2018

classical mechanics - Brachistochrone Problem for Inhomogeneous Potential


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?



Answer




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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