Isochrones and Brachistochrones

Abstract

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.