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This question already has an answer here:

I'm aware that you can always approximate a potential by a quadratic term. But is this the most 'fundamental' reason for the pervasiveness for oscillations?

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marked as duplicate by Qmechanic Jun 30 '18 at 10:11

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Marbles on a hilly surface will tend to roll downhill and settle in the nearby pockets. The reason they stay in the pockets is because the gravitational potential increases in all directions from the bottom of the pocket. A pocket potential can be approximated by a parabolic potential, for small displacements from the bottom of the pocket. The same reasoning applies to other forces. This is why simple harmonic oscillator behavior is so pervasive.

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