Category:Definitions/Abel's Integral Equation

This category contains definitions related to Abel's Integral Equation.
Related results can be found in Category:Abel's Integral Equation.

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 $\map T y$ be a function which specifies the total time of descent for a given starting height.

$\ds \map T {y_0} = \int_{y \mathop = y_0}^{y \mathop = 0} \rd t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\d s} {\d y} \rd y$

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

$\map s y$ is the distance along the curve as a function of height.