I often heard that conservation of momentum is nothing else than Newton's third law.
Ok, If you have only two interacting particles in the universe, this seems to be quite obvious.
However if you have an isolated system of $n$ ($n > 2$) interacting particles (no external forces). Then clearly Newton's third law implies conservation of total momentum of the system. However presuppose conservation of total momentum you only get:
$$
\sum_{i\neq j}^n \mathbf F_{ij} = \frac{d}{d t} \mathbf P = 0
$$
Where $\mathbf F_{ij}$ is the forced acted by the $i$th particle upon the $j$th particle and $\mathbf P$ is the total linear momentum.
But this doesn't imply that $\mathbf F_{ij} = -\mathbf F_{ji}$ for $j \neq i$.
So does conservation of momentum implies Newton's third law in general or doesn't it? Why?
Best Answer
Right, you could satisfy the momentum conservation by forces that don't satisfy "action vs reaction" law $F_{ij}=-F_{ji}$ but the relevant formulae would have to depend on coordinates and momenta of all the particles. If you assume that the particles are controlled by two-body forces only, the momentum conservation does imply that $F_{ij}=-F_{ji}$.