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# Stability Equations

It is evident that if our system is initialized in some equilibrium state, with all of the set to zero, then it will remain in this state for ever. But what happens if the system is slightly perturbed from the equilibrium state?

Let

 (774)

for , where the are small. To lowest order in , the kinetic energy (770) can be written
 (775)

where
 (776)

and
 (777)

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

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

 (778)

where , the are specified in Equation (773), and
 (779)

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
 (780)

Note that, since , the constants weights in the above quadratic form are invariant under interchange of the indices and : i.e.,
 (781)

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

 (782)

for . Note that the above coupled differential equations are linear in the . It follows that the solutions are superposable. Let us search for solutions of the above equations in which all of the perturbed coordinates have a common time variation of the form
 (783)

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
 (784)

which can be written more succinctly as a matrix equation:
 (785)

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.

Next: More Matrix Eigenvalue Theory Up: Coupled Oscillations Previous: Equilibrium State
Richard Fitzpatrick 2011-03-31