4 1. Introduction 0 ya 0 π 2π Figure 1.2. Cycloid for R = 1 2 ya and a = 0 circumference of a circle of radius R rolling along the bottom of the horizontal line y = ya (see Figure 1.2). Huygens (1673, 1986) had previously shown that an inverted cy- cloid is a tautochrone (from ταυτo or τo αυτo, the same, and χρoνoς, time): the time for a heavy particle to fall to the bottom of this curve is independent of the upper starting point. To John Bernoulli’s as- tonishment, the brachistochrone was Huygens’ tautochrone. The brachistochrone is one of many problems where we wish to determine a function, y(x), that minimizes or maximizes the integral J[y(x)] = b a f(x, y(x),y (x)) dx . (1.7) Leonhard Euler first devised a systematic method for solving such problems. In the remainder of this chapter, we will examine three other problems that involve minimizing or maximizing integrals. We will first look at another brachistochrone problem, for travel through the earth. We will then look at the problem of finding the shortest path between two points on some general surface. Finally, we will look at the “soap-film problem,” the problem of minimizing the surface area of a surface of revolution. All of these problems can be attacked using the calculus of variations.

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