# Definition:Abel's Integral Equation

## Definition

Abel's integral equation is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire.

The purpose of Abel's integral equation is to find the shape of the curve into which the wire is bent in order to yield that result:

Let $T \left({y}\right)$ be a function which specifies the total time of descent for a given starting height.

$\displaystyle T \left({y_0}\right) = \int_{y \mathop = y_0}^{y \mathop = 0} \, \mathrm d t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\mathrm d s} {\mathrm d y} \, \mathrm d y$

where:

$y$ is the height of the bead at time $t$
$y_0$ is the height from which the bead is released
$g$ is Acceleration Due to Gravity
$s \left({y}\right)$ is the distance along the curve as a function of height.

## Source of Name

This entry was named for Niels Henrik Abel.

## Historical Note

Niels Henrik Abel devised what is now known as Abel's Integral Equation as a tool by which to solve the Tautochrone Problem in $1823$.