Isochrones and Brachistochrones

Garry J. Tee


Christiaan Huygens proved in 1659 that a particle sliding
smoothly (under uniform gravity) on a cycloid with axis vertically
down reaches the base in a period independent of the starting
point. He built very accurate pendulum clocks with cycloidal
pendulums. Mark Denny has constructed another curve purported to
give descent to the base in a period independent of the starting
point: but the cycloid is the only smooth plane curve with that
property. Johann Bernoulli 1st proved in 1696 that, for any pair
of fixed points, the brachistochrone (the curve of quickest
descent) under uniform gravity is an arc of a cycloid. In 1976,
Ian Stewart asked, what is the brachistochrone for central gravity
under the inverse square law? The solution is found explicitly,
in terms of elliptic integrals.

brachistochrone, quickest descent, constrained motion, central forces, inverse square gravity, elliptic integrals

Math Review Classification
Primary 70D05, 49J15 ; Secondary 01A45, 49-03, 70-03

Last Updated

30 pages

This article is available in: