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I am confused as to how we can assert that energy and momentum are conserved in special relativity. I was attempting a problem on the elastic collision of two particles and came across this.

I have been taught to build up the action in four-space using Lagrangian mechanics and then to relate the expressions from here to the relativistic converted expressions. The method we used was to relate the limiting case of $v<<c$ to classical mechanics with Lagrangian $L=\frac{1}{2}mv^2.$ $\,$[This wiki page outlines what we did essentially; https://en.wikipedia.org/wiki/Four-momentum.]

However, I do not see how this links together... don't we assume that the Lagrangian mechanics is invalid when accounting for relativity? The only thing I can think of is that we assume all 'physics' is true in inertial frames, so is this why we can state energy & momentum are conserved?

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    $\begingroup$ "...don't we assume that the Lagrangian mechanics is invalid when accounting for relativity?" Why do you think this? $\endgroup$ – jacob1729 May 19 at 19:50
  • $\begingroup$ In the same way that classical mechanics is wrong when we account for the fact that light has the same speed in all intertial frames, I thought this was the same for Lagrangian mechanics? $\endgroup$ – user258521 May 19 at 19:51
  • $\begingroup$ The Lagrangian is different, and the specific idea of what a trajectory is is slightly different too I suppose (in Newtonian mechanics one can always parametrise by time, in relativistic mechanics you parametrise paths by their proper time ie arc lengths). But there is still an action principle, which in this case corresponds to maximising the proper time along worldlines. $\endgroup$ – jacob1729 May 19 at 19:55
  • $\begingroup$ So is Lagrangian mechanics a more general framework that is unrelated to the others, in the sense that Newtonian mechanics and relativity have their parallels at low speeds or how quantum mechanics has its Newtonian analogues sometimes? $\endgroup$ – user258521 May 19 at 19:59
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    $\begingroup$ That's a reasonable summary I think. Lagrangian (and Hamiltonian for that matter) mechanics is a framework for all of classical mechanics (i.e. Newtonian and relativistic) which arises in a certain limit from quantum mechanics. $\endgroup$ – jacob1729 May 19 at 20:02
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If the action is invariant under the Poincare group, energy and momentum are conserved by the equations of motion following from it.

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