I'm trying to show the equivalence between the conservation of momentum and the Third Newton's Law.
We have the Newton's third law, that asserts that if object 1 exerts a force $F_{21}$ on object 2, then object 2 always exerts a reaction force $F_{12}$ on object 1 given by $F_{12}$=−$F_{21}$.
On the other hand, we have the principle of conservation of momentum, that asserts that if the net external force $F^{ext}$ on an $N$-particle system is zero, the system's total momentum $P$ is constant.
The proof that the third law implies the principle of conservation of momentum is very easy. Now suppose that we have the principle of conservetion as a true. Take two particles $1$ and $2$. Then if we sum up all the forces in this system we have that if $P$ is the total momentum of the system, then $P^{\prime}=F_{21}+F_{12}+F^{ext}_{1}+F^{ext}_{2}$ where $F^{ext}_{i}$ is the external forces on the particle $i$. Now I'm stuck. I have to suppose that $F_{i}^{ext}=0$ for $i=1,2$? or it is not necessary? Because the Third Law does not requires that the system has no external forces. Can anyone help me?
Best Answer
If your only assumptions are that Newton's first and second laws hold for each particle individually, then you can't prove the third law. The first two laws do not prevent e.g. particle 1 from applying more force on particle 2 than 2 on 1. However, if you require that the first two laws hold for systems of particles (e.g. the rate of change of total momentum of the system is equal to the net external force on the system), then you can show the third law must hold.
The momentum of a system is conserved if and only if the net external force on the system is zero, according to Newton's first law applied to the system. In your simple example, this means $P'=0$ if and only if $F_1^{ext} + F_2^{ext} = 0$. Under these conditions, the third law follows directly.
Even if the net external force is not zero, Newton's second law applied to this system states $P' = F_1^{ext} + F_2^{ext}$. The third law again follows.