In collision problems in special relativity, we often assume conservation of the total energy and momentum. For a single free particle, I understand where these conservation laws come from - the free-particle Lagrangian is independent of the coordinates $x^{\mu}$, which implies the conservation of the corresponding conjugate momenta $p^{\mu}$.

But how can we show that these laws hold for multiple particles? I understand that, because of spacetime symmetries, we will have some sort of energy and momentum quantities which are conserved. But it is unclear to me whether these conserved quantities necessarily have to take the same form as the total energy and momentum we refer to when we do a simple elastic or inelastic collision problem in special relativity (for example, the $E_c$ and $p_c$, the composite energy and momentum, in this problem).

Thanks, let me know if I need to clarify anything!


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