A Tautochrone is a curve described parametrically by the equations $x=a(\theta+\mathrm{sin}\theta)$ and $y=a(1-\mathrm{cos}\theta)$. Here is a graph of the curve for $\theta \in (-2\pi, 2\pi)$

A Tautochrone drawn using SAGE Math

The graph of a tautochrone drawn in SAGE math

This curve is not commonly encountered in mathematics, indeed, I only came across this curve for the first time about two months ago – but it has some very interesting properties. The curve itself is quite simple; however, if a particle were to be released from rest on one of the slopes of the tautochrone, then the time taken for the particle to reach the bottom of the slope is independent of its starting position assuming that the only force acting on it is the gravitational force. In other words, if you were to simultaneously set a ball rolling down the slope from the top of the slope and another ball from halfway down the slope then they would both arrive at the bottom of the slope at exactly the same time.

This property can be proved using energy considerations and some basic trigonometric identities to form the differential equation

$$a\dot{\theta}^{2}\mathrm{cos}^{2}\frac{\theta}{2}=g(\mathrm{sin}^{2}\frac{\theta_{0}}{2}-\mathrm{sin}^{2}\frac{\theta}{2})$$

where $g$ is the gravitational force – and then using integration by substitution we find that the time taken for the particle to reach the bottom is $T=\pi\sqrt{\frac{a}{g}}$ which is independent of starting position – you can download my full, detailed proof of this property here – Basic Properties of the Tautochrone. The derivation of the differential equation and then the integration that follows can all be done using techniques from the A-Level maths and further maths courses.

Since we are assuming that the only force acting on the particle is the gravitational force we can assume that all gravitational potential energy lost (remember that the particle will move down the slope and therefore lose gravitational potential energy) will be converted to kinetic energy. This is the starting point of the whole derivation of the above differential equation and although the resulting differential equation is non-linear, we are fortunate that it is nice enough to be able to solve – non-linear differential equations are notoriously difficult to solve and often impossible to solve analytically.

 

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