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So I know that Newton's third law states that every action has an equal reaction, making a symmetry. But just like how Poincare symmetries form conservation laws, do any Poincare symmetries form Newton's third law?

(Side question: If not, what symmetry is Newton's third law based of?)

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The third law states that momentum is conserved. From Noether we know that momentum consrevation is a consequence of translational symmetry. The tranlations are a subgroup of Poincare. So Yes.

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