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The Universality of 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.

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