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

http://dx.doi.org/10.1090/stml/072/01