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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.