Definition:Abel's Integral Equation

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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.


Also see


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 $1823$.