Chapter 1 Introduction 1.1. The brachistochrone The calculus of variations has a clear starting point. In June of 1696, John (also known as Johann or Jean) Bernoulli challenged the great- est mathematicians of the world to solve the following new problem (Bernoulli, 1696 Goldstine, 1980): Given points A and B in a vertical plane to find the path AMB down which a movable point M must, by virtue of its weight, proceed from A to B in the shortest possible time. Imagine a particle M of mass m, in a vertical gravitational field of strength g, that moves along the curve y = y(x) between the two points A = (a, ya) and B = (b, yb) (see Figure 1.1). The time of descent T of the particle is T = T 0 dt = L 0 dt ds ds = L 0 1 v ds = b a 1 v 1 + y 2 dx , (1.1) where s is arc length, L is the length of the curve, and v is the speed of the particle. If our particle moves without friction, the law of conservation of mechanical energy guarantees that the sum of the particle’s kinetic 1
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