Chapter 5.8, Problem 31P

The Tautochrone. A problem of interest in the history of mathematics is that of finding the Tautochrone-the curve down which a particle will slide freely under gravity alone, reaching the bottom in the same time regardless of its starting point on the curve. This problem arose in the construction of a clock pendulum whose period is independent of the amplitude of its motion. The Tautochrone was found by Christian Huygens in $1673$ by geometrical methods, and later by Leibniz and Jakob Bernoulli using analytical arguments. Bernoulli’s solution (in $1690$) was one of the first occasions on which a differential equation was explicitly solved.

The geometric configuration is shown in Figure $\mathrm{5.8.3}$. The starting point $P\left(a,b\right)$ is joined to the terminal point $\left(0,0\right)$ by the arc $C$. Arc length $s$ is measured from the origin, and $f\left(y\right)$ denotes the rate of change of $s$ with respect to $y$:

$f\left(y\right)=\frac{ds}{dy}={\left[1+{\left(\frac{dx}{dy}\right)}^2\right]}^{1/2}$.         (i)

Then it follows from the principle of conservation of energy that the time $T\left(b\right)$ required for a particle to slide from $P$ to the origin is

$T\left(b\right)=\frac{1}{\sqrt{2g}}{\displaystyle \underset{0}{\overset{b}{\int }}\frac{f\left(y\right)}{\sqrt{b-y}}dy}$.         (ii)

(a) Assume that $T\left(b\right)=T_0$, a constant, for each $b$. By taking the Laplace transform of Eq. (ii) in this case and using the convolution theorem, show that

$F\left(s\right)=\sqrt{\frac{2g}{\pi }}\frac{T_0}{\sqrt{s}}$;         (iii)

Then show that

$F\left(y\right)=\frac{\sqrt{2g}}{\pi }\frac{T_0}{\sqrt{y}}$.         (iv)

Hint: See Problem 37 of Section 5.1.

(b) Combining Eqs. (i) and (iv), show that

$\frac{dx}{dy}=\sqrt{\frac{2\alpha -y}{y}}$,         (v)

where $\alpha =g{T}_0^2/{\pi }^2$.

(c) Use the substitution $y=2\alpha {\sin }^2\left(\theta /2\right)$ to solve Eq. (v), and show that

$x=\alpha \left(\theta +\sin \theta \right),y=\alpha \left(1-\cos \theta \right)$.         (vi)

Equations (vi) can be identified as parametric equations of a cycloid. Thus the Tautochrone is an arc of a cycloid.