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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mathematical modelling reason, yes. Why are branches of the tree oscillating in the wind? The wind will either break them, or there is enough elasticity to oscillate. An existential question. – anna v Jun 30 '18 at 03:46
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1Possible duplicates: https://physics.stackexchange.com/q/159021/2451 , https://physics.stackexchange.com/q/201847/2451 and links therein. – Qmechanic Jun 30 '18 at 04:29
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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.
