Remarks on brachistochrone–tautochrone problems

A number of recent papers have discussed brachistochrone (minimum‐time) and tautochrone (equal‐time) curves for a number of gravitational and rotating systems. The symmetry of the physical system is useful in solving brachistochrone problems expressed in variational form. We point out that such symmetries are equally useful in solving the associated Euler–Lagrange equations, or solving tautochrone differential equations. These brachistochrone problems share a common mathematical form, and therefore a common procedure for reducing the problem to quadrature. It is shown that the cusps characteristic of these curves have a simple physical origin. Finally, the question of the most general spherically symmetric gravitation potential for which the brachistochrone and tautochrone curves are identical is answered in implicit form.