#### BELIEVE ME NOT! **-** **-** A SKEPTIC's GUIDE

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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
-- *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.

Jess H. Brewer -
Last modified: Sun Nov 15 13:51:11 PST 2015