# 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 [1].

**Figure 1.**The contact point

*C*is an instantaneous center of rotation of the rigid wheel, and thus

*v*⊥

*r*.

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**

[1] M. Levi, *Classical Mechanics with Calculus of Variations and Optimal Control*, AMS, Providence, Rhode Island, 2014.