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The third law of Newton assumes that forces are due to the interaction of two bodies. Of course, a particle can interact with several others. But the result of these interactions can always be broken down to interactions between pairs of particles. Is there in nature an interaction, say between three bodies, whose resultant force cannot be decomposed into the sum of forces between the pairs?

I guess that with the introduction of fields to remove the "spooky action at a distance" this problem is solved. But maybe there is some effective interaction for slowly moving particles that is of the form I'm looking for?

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From the perspective of fundamental forces, while electromagnetic and gravitational forces act pairwise between bodies (2-body forces), the strong force can give rise to 3-body interactions. For example the nuclear attraction between the three nucleons in Helium-3 is believed to have 3-body forces involved. You will find the wikipedia article on this topic and its references useful:

https://en.wikipedia.org/wiki/Three-body_force

It should be mentioned though that the strong force isn't well-suited in most (any?) cases for Newtonian style thinking in terms of classical forces due to the magnitude of the coupling constant. Non-perturbative methods in quantum field theory are necessary instead.

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