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
 * $$ \sqrt{(x - x_0\cos\omega t)^2+y^2} - L=0\,\!$$.