---
tag: A56P5S5D6Z
kind: problem
name: Tautochrone problem
msc: [45D05, 01A45]
uses: [G4GWBYTW7B, TDMXFY06A2]
about: [G4GWBYTW7B]
sources:
  - url: https://en.wikipedia.org/wiki/Tautochrone_curve
history:
  posed_by: "Christiaan Huygens"
---

## Problem

A bead slides without friction along a curve in a vertical plane, under a constant downward gravitational acceleration $g > 0$, and starts at rest.
Find the curves with a lowest point that have the following property.
The time the bead needs to reach the lowest point does not depend on its starting point on the curve.

## Remarks

The problem is solved: the answer is an arc of a cycloid with its cusps pointing up.
For the cycloid traced by a circle of radius $a$, the descent time from every starting point is $\pi \sqrt{a / g}$.
Huygens gave a geometric solution and used it for the design of pendulum clocks, in his book Horologium Oscillatorium of 1673.

In the nineteenth century Abel treated the problem through an integral equation.
Let $y$ be the height above the lowest point and $s(y)$ the arc length from the lowest point, and let $\varphi(y) = s'(y)$.
By conservation of energy, the speed at height $y$ of a bead released at height $h$ is $\sqrt{2 g (h - y)}$, so the descent time is
$$T(h) = \frac{1}{\sqrt{2g}} \int_0^h \frac{\varphi(y)}{\sqrt{h - y}} \, dy.$$
Asking that $T(h) = T$ for all $h$ is [[Abel's integral equation|G4GWBYTW7B]] with constant right side $\sqrt{2g} \, T$.
The [[Abel inversion formula|TDMXFY06A2]] gives $\varphi(y) = \frac{\sqrt{2g} \, T}{\pi \sqrt{y}}$, and the curves with $s'(y)$ proportional to $y^{-1/2}$ are exactly the cycloids, with $a = g T^2 / \pi^2$.
Check: on the cycloid $s^2 = 8 a y$, energy conservation gives $s'' = -\frac{g}{4a} s$, a harmonic oscillation whose quarter period $\frac{\pi}{2} \sqrt{4a/g} = \pi \sqrt{a/g}$ does not depend on the amplitude.

## Explanations

### L1 human

On most slides, a ball released higher up takes longer to reach the bottom.
On a cycloid, the steeper upper part speeds up a ball released high just enough to compensate for the longer way.
All balls arrive at the bottom at the same moment, wherever they start.
The problem is a famous early example in which an unknown curve is found from a condition on travel times.
Abel's treatment of it is often seen as a starting point of the theory of integral equations.