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.

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

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

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

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

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

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

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

• Lagrangian mechanics
• Holonomic constraints