# Why are squares, square-roots, and second-order differentials, common in natural laws, but not cubes, cube-roots, and higher order effects? [duplicate]

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?

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