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So, we know from Noether's theorem that the translational symmetry of space (nothing changes if our physical system is located in a different position) implies the conservation of a quantity we call momentum.

We also know that conservation of momentum is equivalent to Newton's third law.

So my question is: Does the physical observation of the homogeneity of space imply Newton's third law? Or are there additional assumptions involved? Does Newton's third law is an unavoidable necessity/consequence in a universe with this symmetry?

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  • $\begingroup$ Newton's third law is most easily understood as a description of contact forces, so it describes a much smaller range of phenomena than Noether's theorem. If you violate the contact force assumption, then you get to problems like in this question physics.stackexchange.com/q/696391 where the physics of fields has to be severely limited. In general I would therefor not say that the third law is "equivalent" to momentum conservation because it's much more limited in its scope than what momentum conservation applies to. $\endgroup$ Apr 23, 2023 at 22:13
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/12122/2451 and links therein. $\endgroup$
    – Qmechanic
    Apr 24, 2023 at 1:38

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Does the physical observation of the homogeneity of space imply Newton's third law? Or are there additional assumptions involved?

There are two additional assumptions beyond the observed symmetry of spatial homogeneity.

First is that the laws of physics can be written in the form of a Lagrangian. There is, in principle, no reason that the laws of physics must be based on an action principle. If that were not the case then Noether’s theorem would not apply.

Second is that there are objects that interact pairwise via mechanical forces. With those two additional assumptions Newton’s third law is inevitable. The forces between objects must obey Newton’s third law.

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