Relaxing the assumption of equal masses

Above we have assumed that each degree of freedom has an equal effective mass. We now relax this assumption, using Lagrange s equation of motion, 

For a system of point masses, each described by Cartesian coordinates, Lagrange s equation reduces to Newton s second law of motion. For small departures about a minimum energy state, Lagrange s eqnations of motion typically generate a system of the linearized form 

We thus must perform normal mode analysis on the matrix M //, where 

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