Edit in response to @GennaroTedesco 's answer For an isolated system consisting of $N$-particles, its total linear momentum is conserved. In particular, for a system of two colliding particles i.e., $N=2$, the conservation of momentum implies $$\textbf{p}_1+\textbf{p}_2=\textbf{p}^\prime_1+\textbf{p}^\prime_2$$ where $\textbf{p}_1,\textbf{p}_2$ are the individual momenta of the particles before coliision and $\textbf{p}^\prime_1,\textbf{p}^\prime_2$ are those after the collision. In the centre-of-mass frame $\textbf{p}_1+\textbf{p}_2=\textbf{p}^\prime_1+\textbf{p}^\prime_2=0$ Therefore, this is useful in analysing why the process $e^++e^-\to \gamma$ is not possible but $e^++e^-\to \gamma+\gamma$ is possible.

Are there similar situations where the conservation of linear momentum can be used to kinematically forbid some process in a system with $N>2$ (say, $N=3$) particles?


closed as too broad by Emilio Pisanty, John Rennie, Kyle Kanos, Jon Custer, David Hammen Jul 13 '17 at 11:32

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    $\begingroup$ What do you mean by "how useful"? The conservation laws are useful because they work. $\endgroup$ – Steeven Jul 12 '17 at 14:55
  • $\begingroup$ @Steeven You can see the edited question. I explained "how useful" it is when $N=2$. $\endgroup$ – SRS Nov 6 '18 at 16:21

Conservation laws are nothing but equations of motion that are solved for free, namely $$ f(q,p;t) = f_0 $$ therefore they reduce the degrees of freedom of the whole system, namely the collection of remaining equations you have to solve in order to have the complete solution. If the system is integrable, then those conserved quantities can be inverted back to express the $(q,p)$ in terms of some initial quantities $(c_0,c_1,\dots,c_n)$. In the case of the two-body problem, the conservation of angular momentum allows to reduce the dimentionality of the spatial manifold where the motion occurs, therefore making the integration of the remaining equation of motion easier.

This said, on a more abstract level, they may provide information on the symmetric and algebraic structure of the bundle where the mathematics of the motion lives (see the case of Bianchi identity for the electromagnetic field or any other gauge field).


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