Let

(774) |

where

and

Note that the weights in the quadratic form (775) are now

Taylor expanding the potential energy function about the equilibrium state, up to second-order in the , we obtain

Now, we can set to zero without loss of generality. Moreover, according to Equation (773), the are all zero. Hence, the expression (778) reduces to

Note that, since , the constants weights in the above quadratic form are invariant under interchange of the indices and :

With and specified by the quadratic forms (775) and (780), respectively, Lagrange's equations of motion (772) reduce to

for . Now, Equations (782) are a set of second-order differential equations. Hence, the most general solution contains arbitrary constants of integration. Thus, if we can find sufficient independent solutions of the form (783) to Equations (782) that the superposition of these solutions contains arbitrary constants then we can be sure that we have found the most general solution. Equations (782) and (783) yield

which can be written more succinctly as a matrix equation:

Here, is the real [see Equation (779)] symmetric [see Equation (781)] matrix of the values. Furthermore, is the real [see Equation (770)] symmetric [see Equation (777)] matrix of the values. Finally, is the vector of the values, and is a null vector.