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

A simple 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.

File:Pendulum02.JPG

A simple pendulum with oscillating pivot point

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

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

See also

• Lagrangian mechanics
• Holonomic constraints