Isochrones and Brachistochrones
by
Garry J. Tee
University of Auckland

Christiaan Huygens greatly advanced mathematics, physics and technology in his book "Horologium Oscillatorum" (1673). He shewed that a particle sliding smoothly under uniform gravity, on a cycloid with its axis vertically downwards, oscillates isochronously, in a period independent of amplitude; and so he called the cycloid the ïsochrone". Huygens constructed very accurate clocks, in which the pendulum was constrained to move on a cycloid.

Mark Denny has recently constructed another curve, purported to have that same isochronous property. But, it is not difficult to prove that every curve giving isochronous oscillations, under uniform gravity, is a cycloid with its axis vertically downwards.

The calculus of variations was founded by Johann Bernoulli 1st in 1696, when he found that the the curve of quickest descent between any pair of points, under uniform gravity, is an arc of a cycloid with axis vertically downwards. Thus, under uniform gravity, the cycloid is the brachistochrone as well as the isochrone.

Ian Stewart asked whether the brachistochrone was known for inverse square gravity and Henry Forder cited E. J. Routh's treatment in ``A Treatise on Dynamics of a Particle" (1898). Routh's treatment of brachistochrones for central forces is very sketchy, with no case worked out fully. But, for inverse square gravity, the brachistochrone can be constructed in terms of elementary functions.

Date received: June 14, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabd-55.