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I just happened to reading Goldstein's "Classical Mechanics" last
night. This is the classic brachistochrone problem of variational
calculus:
A mass moves from point A to point B under the influence of a uniform
gravitational field (and a reaction force constraining the mass to move
along a nonballistic path). What path does the mass take to minimize
the time in going from A to B?
The answer is a cycloid, which is not the same as a straight line, the
path which minimizes the path length from A to B. A cycloid is a
"roller coaster valley"-shaped path (imagine the path taken by a point
on a rolling circle).
I seem to remember solving the brachistochrone problem in my advanced
calculus class except we assumed a radially symmetric gravitational
field. The problem posed was: A frictionless high-speed train travels
only under the influence of gravity in a tunnel connecting two points on
the earth's surface. What shape must the tunnel have to minimize the
travel time of the train? My recollection is that the answer is not a
straight line either, but, rather, a hypocycloid (imagine the path taken
by a point on a small circle rolling inside a large circle). Since the
tunnel is inside the earth, the gravitational force on the train is
proportional to the train's distance from the center of the earth.
Note: The brachistochrone for central gravity under the inverse square
law was not found until 1976 (!). The solution by Ian Stewart is found
explicitly, in terms of elliptic integrals. (Tee, Garry J., "Isochrones
and Brachistochrones," University of Auckland, Auckland, New Zealand,
1998. Available at
http://www.math.auckland.ac.nz/Research/DeptRep/405.html)
Glenn A. Carlson
Not really surprising at all in the limit of a very gradual slope.
John