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I hope my question have captured concisely what I am asking about.

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    $\begingroup$ I am wondering why symmetry plus superposition are not sufficient for you - without invoking N3. If you have two spherical objects in line, what direction would you like the force to be if not along the line joining them? Symmetry would seem to dictate it (what other direction can the force pick?) $\endgroup$ – Floris Aug 5 '16 at 12:04
  • $\begingroup$ That seem's circular to me. Had we not discovered laws that resemble symmetry, we wouldn't be as comfortable using the concept of symmetry to dig out intuition from the way things work in nature. I am open to being convinced otherwise. $\endgroup$ – kfs Aug 5 '16 at 12:08
  • $\begingroup$ Alright - abandon all prior knowledge of symmetry. What direction would your "non central" force want to be? You need some external reference - or accept that the world is a very random place, where there is no guarantee that A leads to B. $\endgroup$ – Floris Aug 5 '16 at 12:18
  • $\begingroup$ the magnetic field does not produce a central force $\endgroup$ – user126422 Aug 5 '16 at 12:25
  • $\begingroup$ Related: physics.stackexchange.com/q/12122/2451 , physics.stackexchange.com/q/16162/2451 and links therein. $\endgroup$ – Qmechanic Aug 5 '16 at 12:27
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Newton's third law is basically just conservation of linear momentum, and it arises from a fundamental symmetry called spatial shift symmetry using Noether's theorem. This symmetry basically says that if we move the system we ar looking at by some distance in space then the physics is not affected. The actual statement is that the action is invarient under spatial translations. see Conservation of Momentum for a discussion of what this means.

A central force means that the force is spherically symmetric, and this is also a symmetry. Noether's theorem tells us that this symmetry also gives rise to a conservation law but this time it is the conservation of angular momentum rather than linear momentum. So if you have a system where angular momentum is conserved that means any forces must be central.

So no, Newton's third law isn't directly related to the existance or otherwise of a central force, but there is a connection in that both the third law and the existance of a central force are related to conservation of momentum.

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  • $\begingroup$ Does a central force then imply conservation of (angular) momentum of the system? $\endgroup$ – kfs Aug 5 '16 at 15:53
  • $\begingroup$ @kfs: yes, it does. $\endgroup$ – John Rennie Aug 5 '16 at 15:55
  • $\begingroup$ Is it, at all, sensible to ask which is more fundamental? $\endgroup$ – kfs Aug 5 '16 at 15:57
  • $\begingroup$ @kfs: Noether's theorem links conservation laws to summetries, so either one implies the other. Which is more fundamental is a matter of personal opinion, though most physicists I know tend to regard the symmetry as fundamental. For example we regard special relativity as arising from Lorentz covariance and general relativity from diffeomorphism invariance. $\endgroup$ – John Rennie Aug 5 '16 at 16:07

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