Applied Mathematics/Lagrange Equations

Lagrange Equation

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where  
The equation above is called Lagrange Equation.

Let the kinetic energy of the point mass be   and the potential energy be  .
  is called Lagrangian. Then the kinetic energy is expressed by

 
 

Thus

 
 

Hence the Lagrangian   is

 
 
 

Therefore   relies on only   and  .   relies on only   and  . Thus

 
 

In the same way, we have