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.

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.