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Quadratic potential is a good approximation for the minima of smooth functions. This is I think is the reason that simple harmonic motion is so common.

What is the reason behind this?

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    $\begingroup$ It's the simplest polynomial with a stable minimum. $\endgroup$ – Demosthene Feb 22 '17 at 8:55
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    $\begingroup$ Just as a suggestion, there a lot of wonderful effects that arises when you go further with your approximation, not stopping at the quadratic term. Take a look, for instance, to the second harmonic generation or the electrooptic effects in nonlinear optics! $\endgroup$ – JackI Feb 22 '17 at 9:06
  • $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/159021/2451 and links therein. $\endgroup$ – Qmechanic Feb 22 '17 at 9:59
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    $\begingroup$ Possible duplicate of Why is the harmonic oscillator so important? $\endgroup$ – Brian Moths Feb 22 '17 at 13:08
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    $\begingroup$ In 3D, it is the only possible divergence-free force field originating from a pointlike monopol source. $\endgroup$ – peterh says reinstate Monica Feb 22 '17 at 18:39