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Recently, I have learnt about the central force motion. I came across some forces with different distance dependence, for example, k/r^2(common) one, k/r^4, or even the combined one k/r^2 + k'/r^4 (k, k' are force constant).

I remembered when I first learnt the inverse square law, I was convinced that the inverse square law came from the density of the field line on a surface. However, how the forces with 1/r^3, 1/r^4 exists? Can they be derived by classical means?

I have searched for different information but I cannot get a good answer. I hope someone can help me, or relevant reference will do.

Thank you!

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    $\begingroup$ Nothing stops you from inserting any kind of force law into the equations of classical mechanics. There is no "selection mechanism" that would chose one over the other. Having anything but a 1/r potential has very important (and unfortunate) implications for orbital motion, though, see en.wikipedia.org/wiki/Bertrand%27s_theorem. If you have systems with many degrees of freedom, effective potentials can also take on e.g. Yukawa or van der Walls form etc.. Once you move on to (quantum) field theory, things get dicy... only very few models turn out to be self-consistent. $\endgroup$ – CuriousOne Mar 27 '16 at 4:44
  • $\begingroup$ Can I say that most of the time, we need to observe the orbits to determine the force law? $\endgroup$ – tomchan516 Mar 27 '16 at 6:09
  • $\begingroup$ In order to deduce the force law one would have to observe all possible orbits, so if you can do that... yes, but that's usually not possible in practice. Just take the case of a planet, where the density distribution causes a non-trivial gravitational potential. That potential can not be measured with orbits because one can't fly a tiny satellite around inside a planet. $\endgroup$ – CuriousOne Mar 27 '16 at 9:03
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While electric forces are 1/R^2, that's only between two charges. If you look at distributions of mixed charges, the force laws include dipole, quadrupole, octopole... and while those are a bit more complex than simple central forces (because they depend on orientation), the scaling with distance includes 1/R^N terms (N being two or more).

Yukawa's potential for the strong atomic force came from an insight, that central forces were mediated by virtual particles (the photon, for electric force), and those particles made the Faraday-style 'lines of force' picture into a special case that depicts 1/R^2 forces due to massless virtual particles. He then calculated the mass of the pion, based on the known properties of nuclear binding, and on the uncertainty principle linking mass to lifetime. If that insight is correct, all fundamental forces in 3-dimensional space are likely to be inverse square law with an (optional) exponential decay factor.

Yukawa's particle, the pion, was found. His insight seems correct, so far.

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