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