Is the tautochrone curve unique?

TL;DR

This paper demonstrates that the tautochrone problem admits infinitely many solutions beyond the classical cycloid, by exploring potential energies and track shapes that produce identical oscillation periods.

Contribution

It reveals the existence of infinitely many tautochrone curves and potential functions, extending the classical solution and providing a comprehensive analysis of periodic motions under various potentials.

Findings

Infinite tautochrone curves exist beyond the cycloid.

Multiple potential energies can produce the same oscillation period.

There are infinitely many tracks leading to the same period of oscillation.

Abstract

The answer to this question is no. In fact, in addition to the solution first obtained by Christiaan Huygens in 1658, given by the cycloid, we show that there is an infinite number of tautochrone curves. With this goal, we start by briefly reviewing an the problem of finding out the possible potential energies that lead to periodic motions of a particle whose period is a given function of its mechanical energy. There are infinitely many solutions, called sheared potentials. As an interesting example, we show that a P\"oschl-Teller and the one-dimensional Morse potentials are sheared relative to one another for negative energies, clarifying why they share the same periods of oscillations for their bounded solutions. We then consider periodic motions of a particle sliding without friction over a track around its minimum under the influence of an uniform gravitational field. After a brief historical survey of the tautochrone problem we show that, given the period of oscillations, there is an infinity of tracks which lead to the same periods. As a bonus, we show that there are infinitely many tautochrones.