1.2. The terrestrial brachistochrone 5 1.2. The terrestrial brachistochrone History repeats itself. In August of 1965, Scientific American pub- lished an article on “High-Speed Tube Transportation” (Edwards, 1965). Edwards proposed tube trains that would fall through the earth, pulled by gravity and helped along by pneumatic propulsion. The advantages cited by Edwards included: (1) It brings most of the tunnel down into deep bedrock, where the cost of tunneling by blasting or by boring is reduced and incidental earth shifts are minimized the rock is more homogeneous in con- sistency and there is less likelihood of water inflow. (2) The nuisance to property owners decreases with depth, so the cost of easements should be lower. (3) A deep tunnel does not interfere with subways, building foundations, utilities, or water wells. . . . (4) The pendulum ride is uniquely comfortable for the passenger. . . . Lest you think this pure fantasy, a pneumatic train was con- structed in New York City, under Broadway, from Warren Street to Murray Street, in 1870 by Alfred Ely Beach (an early owner of Scien- tific American). This was New York City’s first subway (Roess and Sansone, 2013). You can see a drawing of the pneumatic train on the wallpaper in older Subway Sandwich shops. Cooper (1966a) then pointed out that straight-line chords lead to needlessly long trips through the earth. He used the calculus of vari- ations to derive a differential equation for the fastest tunnels through the earth and integrated this equation numerically. Venezian (1966), Mallett (1966), Laslett (1966), and Patel (1967) then found first in- tegrals and analytic solutions for this problem. See Cooper (1966b) for a summary. Let us take a closer look at this terrestrial brachistochrone prob- lem. Assume that the earth is a homogeneous sphere of radius R. Consider a section through the earth with polar coordinates centered at the heart of the earth (see Figure 1.3). Imagine a particle of mass
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