Rheonomous
# Rheonomous

A mechanical system is rheonomous if its equations of constraints contain the time as an explicit variable. Such constraints are called rheonomic constraints. The opposite of rheonomous is scleronomous.

## Example: simple 2D pendulum

As shown at right, a simple pendulum is a system composed of a weight and a string. The string is attached at the top end to a pivot and at the bottom end to a weight. Being inextensible, the string has a constant length. Therefore, this system is scleronomous; it obeys the scleronomic constraint

    ${\sqrt {x^{2}+y^{2}}}-L=0\,\!$,

where $(x,\ y)\,\!$ is the position of the weight and $L\,\!$ the length of the string.

The situation changes if the pivot point is moving, e.g. undergoing a simple harmonic motion

    $x_{t}=x_{0}\cos \omega t\,\!$,

where $x_{0}\,\!$ is the amplitude, $\omega \,\!$ the angular frequency, and $t\,\!$ time.

Although the top end of the string is not fixed, the length of this inextensible string is still a constant. The distance between the top end and the weight must stay the same. Therefore, this system is rheonomous; it obeys the rheonomic constraint

    ${\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!$.
