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The Universality of ${\cal SHM}$

Why is SHM characteristic of such an enormous variety of phenomena? Because for sufficiently small displacements from equilibrium, every system with an equilibrium configuration satisfies the first condition for SHM: the linear restoring force. Here is the simple argument: a linear restoring force is equivalent to a potential energy of the form   $V(q) = \onehalf k \, q^2$  -- i.e. a ``quadratic minimum'' of the potential energy at the equilibrium configuration  q = 0. But if we ``blow up'' a graph of  V(q)  near  q = 0,  every minimum looks quadratic under sufficient magnification! That means any system that has an equilibrium configuration also has some analogue of a ``potential energy'' which is a minimum there; if it also has some form of inertia so that it tends to stay at rest (or in motion) until acted upon by the analogue of a force, then it will automatically exhibit SHM for small-amplitude displacements. This makes SHM an extremely powerful paradigm.