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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?

Peter Mortensen
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veronika
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    It's the simplest polynomial with a stable minimum. – Demosthene Feb 22 '17 at 08:55
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    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! – JackI Feb 22 '17 at 09:06
  • Essentially a duplicate of https://physics.stackexchange.com/q/159021/2451 and links therein. – Qmechanic Feb 22 '17 at 09:59
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    Possible duplicate of Why is the harmonic oscillator so important? – Brian Moths Feb 22 '17 at 13:08
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    In 3D, it is the only possible divergence-free force field originating from a pointlike monopol source. – peterh Feb 22 '17 at 18:39
  • @peterh Write an answer expanding on this and I will accept it – veronika Feb 22 '17 at 18:41
  • @Demosthene I think your answer is very good from the Occam's razor view, but I think the selecting only the polynomial functions is an arbitrary choice from the viewpoint of the question. – peterh Feb 22 '17 at 18:41
  • @veronika I can't, it is already held. I can vote to reopen it, and I do, but my vote will probably not enough. Also you can vote to reopen your own question, so we will need only 3 (unfortunately, probably there won't be, but we can try). – peterh Feb 22 '17 at 18:48
  • @veronika Furthermore, your question is about the potential of things in a linear force field (i.e. $F=c \cdot r$), my comment is about the force around pointlike charges. Thus my comment doesn't answer your question. But you can ask a new question about that why are there always $\frac{1}{x^2}$-like forces. – peterh Feb 22 '17 at 18:53
  • @peterh Of course, of course, but the question is about the quadratic approximation around minima. The Taylor series of most usual potentials (periodic functions, exponentials, power laws, etc.) will always be polynomial. – Demosthene Feb 23 '17 at 09:02

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