Applied Mathematics/Lagrange Equations

Lagrange EquationEdit

 

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