## Tautochrone Problem

Find the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a Cycloid, a fact first discovered and published by Huygens in Horologium oscillatorium (1673). Huygens also constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in an arc of a Cycloid.

The parametric equations of the Cycloid are

 (1) (2)

To see that the Cycloid satisfies the tautochrone property, consider the derivatives
 (3) (4)

and
 (5)

Now
 (6)

 (7)

 (8)

so the time required to travel from the top of the Cycloid to the bottom is
 (9)

However, from an intermediate point ,
 (10)

so
 (11)

Now let
 (12) (13)

so
 (14)

and the amount of time is the same from any point!

References

Muterspaugh, J.; Driver, T.; and Dick, J. E. The Cycloid and Tautochronism.'' http://php.indiana.edu/~jedick/project/intro.html.

Muterspaugh, J.; Driver, T.; and Dick, J. E. P221 Tautochrone Problem.'' http://php.indiana.edu/~jedick/project/project.html.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 54-60 and 384-385, 1991.