Brachistochrones Under Central Forces
by
Garry J. Tee
University of Auckland

Johann Bernoulli 1st proved in 1696 that the brachistochrone (curve of quickest descent) under uniform gravity is the cycloid. The brachistochrones have now been generalized to general central forces, both attractive and repulsive, with full details for inverse square forces and for logarithmic potential.

With any smooth potential, as the arclength of a brachistochrone converges to 0, its shape converges to that of a cycloid. Thus, for repulsive central force, each sufficiently small brachistochrone is bounded in radius; but an unbounded straight line radiating from the startpoint is also a brachistochrone, and so some brachistochrones are unbounded. The bounded and unbounded brachistochrones are separated by a critical brachistochrone, which is bounded in radius.

For inverse square forces the brachistochrones are constructed in terms of elliptic integrals – except that (for repulsion) the critical brachistochrone is constructed in terms of elementary functions.

Date received: September 23, 1999