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 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
x 2 + y 2 − L = 0 {\displaystyle {\sqrt {x^{2}+y^{2}}}-L=0\,\!} , where ( x , y ) {\displaystyle (x,\ y)\,\!} is the position of the weight and L {\displaystyle 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 ω t {\displaystyle x_{t}=x_{0}\cos \omega t\,\!} , where x 0 {\displaystyle x_{0}\,\!} is the amplitude, ω {\displaystyle \omega \,\!} the angular frequency, and t {\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
( x − x 0 cos ω t ) 2 + y 2 − L = 0 {\displaystyle {\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!} .
See also Lagrangian mechanics Holonomic constraints
References