SIAM News Blog
SIAM News

# A Cycloid is a Tautochrone

#### A Short Proof

In 1659 Christiaan Huygens answered a question that came to him as he watched a swinging chandelier in a church: What curve has the property that a bead sliding along it under uniform gravity and with no friction will oscillate with a period independent of the amplitude?

The answer turned out to be the cycloid generated by a circle, as illustrated in Figure 1. Several solutions to this problem have been found; Abel’s is particularly remarkable . Figure 1. The contact point C is an instantaneous center of rotation of the rigid wheel, and thus vr.

Presented here is a very short geometrical proof of the tautochronous property of the cycloid. It is based on the fact that $$\bf{v} \perp \bf{r}$$, as explained in Figure 1.1

To prove that the cycloid is a tautochrone, it suffices to show that the arclength distance $$s$$ from the bottom of the cycloid behaves as a harmonic oscillator:

$\begin{equation} a = −ks, \end{equation}$

where $$a = \ddot{s}$$, for some constant $$k$$. (This idea, which I had learned from Henk Broer, is attributed to Lagrange.) Because $$a = 0$$ when $$s = 0$$, we just need to verify that

$\begin{equation}\tag{1} da = −k ds. \end{equation}$

But $$da = d(g ~\mathrm{cos}~ \theta) = −g ~\mathrm{sin}~ \theta~ d\theta$$. And from Figure 2 we have $$ds = D sin \theta d \theta$$. Comparing these expressions for $$da$$ and $$ds$$ proves (1) with $$k = g/D$$. QED Figure 2. Proof that a = –ks, using the fact that PC'PC.

1 Incidentally, building on this fact, the line of velocity of every point on a rolling wheel (in the ground reference frame) passes through the topmost point of the wheel. A pebble stuck to the tire always aims straight at, or straight away from, the topmost point of the wheel!

References
 M. Levi, Classical Mechanics with Calculus of Variations and Optimal Control, AMS, Providence, Rhode Island, 2014.

Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.