Normal Coordinates in Classical Mechanics

The potential energy V depends on the mutual positions of the nuclei and therefore upon the coordinates If we restrict 

Using the coordinates g , we now set up the classical equations of motion in the Lagrangian form (Sec. lc). In this case the kinetic energy T is a function of the velocities g only, and the potential energy V is a function of the coordinates q only, and in consequence the Lagrangian equations have the form 

On introducing the above expressions for T and V we obtain the equations of motion 

In case that the potential-energy function involves only squares q] and no cross-products g.g,- with i j that is, if 6 , vanishes for i j, then these equations of motion can be solved at once. They have the form 

Now it is always possible by a simple transformation of variables to change the equations of motion from the form 37-8 to the form 37-9 that is, to eliminate the cross-products from the potential energy and at the same time retain the form 37-3 for the kinetic energy. Let us call these new coordinates Qi(l = 1, 2, , 3n). In terms of them the kinetic and the potential energy would be written 

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