Gauss' principle of least constraint | Wikipedia audio article
Channel: wikipedia tts
Date: 30/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Gauss%27s_principle_of_least_constraint


00:03:10 1 Hertz's principle of least curvature
00:07:50 2 See also



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SUMMARY
=======
The principle of least constraint is another formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829.
The principle of least constraint is a least squares principle stating that the true accelerations of a mechanical system of 
  
    
      
        N
      
    
    {\displaystyle N}
   masses is the minimum of the quantity

  
    
      
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    {\displaystyle Z\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{N}m_{k}\left|{\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}-{\frac {\mathbf {F} _{k}}{m_{k}}}\right|^{2}}
  for all accelerations 
  
    
      
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    {\displaystyle \{{\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}\}}
   satisfying the imposed constraints, where 
  
    
      
        
          m
          
            k
          
        
      
    
    {\displaystyle m_{k}}
  , 
  
    
      
        
          
            r
          
          
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    {\displaystyle \mathbf {r} _{k}}
   and 
  
    
      
        
          
            F
          
          
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    {\displaystyle \mathbf {F} _{k}}
   represent the mass, position and applied non-constraint forces of the 
  
    
      
        
          
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    {\displaystyle \mathrm {k^{th}} }
   mass.
Note that the set of accelerations satisfying the imposed constraints is in general dependent on the current state of the system,

  
    
      
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