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