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Generalization of ${\cal SHM}$

As for all the other types of equations of motion, SHM need not have anything to do with masses, springs or even Physics. Even within Physics, however, there are so many different kinds of examples of SHM that we go out of our way to generalize the results: using ``q'' to represent the ``coordinate'' whose displacement from the equilibrium ``position'' (always taken as  q = 0) engenders some sort of restoring ``force''   $Q = - k \, q$  and ``$\mu$'' to represent an ``inertial factor'' that plays the rôle of the mass, we have

$$\ddot{q} = - \left( k \over \mu \right) q$$ (13.26)

for which the solution is the real part of

$$q(t) = q_0 \, e^{i \, \omega \, t} \qquad \hbox{\rm where} \qquad \omega = \sqrt{ k \over \mu }$$ (13.27)

When some form of ``drag'' acts on the system, we expect to see the qualitative behaviour pictured in Fig. 13.5 and described by Eqs. (21) and (22). Although one might expect virtually every real example to have some sort of frictional damping term, in fact there are numerous physical examples with no damping whatsoever, mostly from the microscopic world of solids.