Lecture 1
The Brachistochrone
Although the roots of the calculus of variations can be traced to muc
earlier times, the birth date of the subject is widely considered to b
June of
1696.1
That is when John Bernoulli posed the celebrate
problem of the brachistochrone or curve of quickest descent, i.e., t
determine the shape of a smooth wire on which a frictionless bea
slides between two fixed points in the shortest possible time.
x
0 0.5 1
y
0
0.5
1
Figure 1.1. A frictionless bead on a wire.
For the sake of definiteness, let us suppose that the points i
question have coordinates (0, 1) and (1, 0), and that the bead slide
1See,
e.g., Bliss [5, pp. 12-13 and 174-179] or Hildebrandt & Tromba [21, pp. 2
27 and 120-123], although Goldstine [17, p. vii] prefers the earlier date of 1662 whe
Fermat applied his principle of least time to light ray refraction.
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