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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?

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  • $\begingroup$ One well-known beam bending theory uses $4^{th}$ order derivatives. $\endgroup$
    – Gert
    May 8 '20 at 19:44
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    $\begingroup$ Kepler’s 3rd law is another nice counter-example. $\endgroup$
    – G. Smith
    May 8 '20 at 19:55
  • $\begingroup$ 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? $\endgroup$
    – fra_pero
    May 8 '20 at 19:59
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/162883/2451 , physics.stackexchange.com/q/226994/2451 and links therein. $\endgroup$
    – Qmechanic
    May 8 '20 at 20:02

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