18 1. Introduction minimal surface equation. (Meusnier, like Lagrange, seemed unaware of Euler’s earlier analysis of the catenoid.) The study of minimal surfaces has grown to become one of the richest areas of mathematical research. In the remainder of this book, we will look at many other prob- lems in the calculus of variations. 1.5. Recommended reading Goldstine (1980), Fraser (2003), Kolmogorov and Yushkevich (1998), and Kline (1972) provide useful historical surveys of the calculus of variations. Icaza Herrera (1994), Sussmann and Willems (1997), and Stein and Weichmann (2003) have written stimulating historical articles about the brachistochrone problem. An experimental study of the brachistochrone (using a “Hot Wheels” car) was carried out by Phelps et al. (1982). The original 1697 solutions of John and Jacob Bernoulli can be found, translated into English, in Struik (1969). John Bernoulli’s solution was recently reviewed by Erlichson (1999) and reviewed and generalized by Filobello-Nino et al. (2013). If the endpoints A and B lie above the surface of the earth, but at vastly different heights, the gravitational field is no longer constant. One must instead determine the curve of swiftest descent in an at- tractive, inverse-square, gravitational field. This problem has been discovered repeatedly. Recent treatments include those of Singh and Kumar (1988), Parnovsky (1998), Tee (1999), and Hurtado (2000). Goldstein and Bender (1986) analyzed the brachistochrone in the presence of relativistic effects and Farina (1987) showed that John Bernoulli’s optical method can also be used to solve this relativis- tic problem. Kamath (1992) determined the relativistic tautochrone using fractional calculus. The idea of high-speed tunnels through the earth is quite old. In Lewis Carroll’s (1894) Sylvie and Bruno Concluded, Mein Herr describes a system of railway trains, without engines, powered by gravity:

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