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Natural laws often feature squares and square roots, and second-order differential equations. Cubic laws, cube-roots, and third-order differentials are fairly rare.

(Some counter-examples: square-cube laws turn up when area/volume effects are scaled, and Stefan–Boltzmann law involves a fourth-power. Perhaps I'm just ignorant but I struggle to come up with many more.)

Is there a deep reason why higher-order effects would be rarer?

Qmechanic
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spraff
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  • One well-known beam bending theory uses $4^{th}$ order derivatives. – Gert May 08 '20 at 19:44
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    Kepler’s 3rd law is another nice counter-example. – G. Smith May 08 '20 at 19:55
  • Does your question boil down to: Why gravitational force and electrostatic force (i.e. the fundamental long-range forces) fall down as the inverse distant square? – fra_pero May 08 '20 at 19:59
  • Possible duplicates: https://physics.stackexchange.com/q/162883/2451 , https://physics.stackexchange.com/q/226994/2451 and links therein. – Qmechanic May 08 '20 at 20:02

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