Rheonomous

<h2 id="example-simple-2d-pendulum">Example: simple 2D pendulum</h2>

<p>As shown at right, a simple <a href="/facts/Pendulum/D2hQXwlI">pendulum</a> 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
</p>

x
              
                2
              
            
            +
            
              y
              
                2
              
            
          
        
        −
        L
        =
        0
        
        
      
    
    {\displaystyle {\sqrt {x^{2}+y^{2}}}-L=0\,\!}
  
,
<p>where 
  
    
      
        (
        x
        ,
         
        y
        )
        
        
      
    
    {\displaystyle (x,\ y)\,\!}
  
 is the position of the weight and 
  
    
      
        L
        
        
      
    
    {\displaystyle L\,\!}
  
 the length of the string.
</p>

<p>The situation changes if the pivot point is moving, e.g. undergoing a <a href="/facts/Simple_harmonic_motion/u2cGAd91">simple harmonic motion</a>
</p>

x
          
            t
          
        
        =
        
          x
          
            0
          
        
        cos
        ⁡
        ω
        t
        
        
      
    
    {\displaystyle x_{t}=x_{0}\cos \omega t\,\!}
  
,
<p>where 
  
    
      
        
          x
          
            0
          
        
        
        
      
    
    {\displaystyle x_{0}\,\!}
  
 is the amplitude, 
  
    
      
        ω
        
        
      
    
    {\displaystyle \omega \,\!}
  
 the angular frequency, and 
  
    
      
        t
        
        
      
    
    {\displaystyle t\,\!}
  
 time.
</p><p>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
</p>

(
            x
            −
            
              x
              
                0
              
            
            cos
            ⁡
            ω
            t
            
              )
              
                2
              
            
            +
            
              y
              
                2
              
            
          
        
        −
        L
        =
        0
        
        
      
    
    {\displaystyle {\sqrt {(x-x_{0}\cos \omega t)^{2}+y^{2}}}-L=0\,\!}
  
.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian mechanics</a></li>
<li><a href="/facts/Holonomic_constraints/cvxCH0Ar">Holonomic constraints</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). United States of America: Addison Wesley. p. 12. ISBN 0-201-02918-9. Constraints are further classified according as the equations of constraint contain the time as an explicit variable (rheonomous) or are not explicitly dependent on time (scleronomous). <a href="0-201-02918-9" target="_blank">0-201-02918-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Spiegel, Murray R. (1994). Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory. Schaum's Outline Series. McGraw Hill. p. 283. ISBN 0-07-060232-8. In many mechanical systems of importance the time t does not enter explicitly in the equations (2) or (3). Such systems are sometimes called scleronomic. In others, as for example those involving moving constraints, the time t does enter explicitly. Such systems are called rheonomic. <a href="0-07-060232-8" target="_blank">0-07-060232-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). United States of America: Addison Wesley. p. 12. ISBN 0-201-02918-9. Constraints are further classified according as the equations of constraint contain the time as an explicit variable (rheonomous) or are not explicitly dependent on time (scleronomous). <a href="0-201-02918-9" target="_blank">0-201-02918-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Spiegel, Murray R. (1994). Theory and Problems of THEORETICAL MECHANICS with an Introduction to Lagrange's Equations and Hamiltonian Theory. Schaum's Outline Series. McGraw Hill. p. 283. ISBN 0-07-060232-8. In many mechanical systems of importance the time t does not enter explicitly in the equations (2) or (3). Such systems are sometimes called scleronomic. In others, as for example those involving moving constraints, the time t does enter explicitly. Such systems are called rheonomic. <a href="0-07-060232-8" target="_blank">0-07-060232-8</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Rheonomous open-in-new

Rheonomous