# Exact statement of conservation of momentum

According to Wikipedia, the conservation of momentum states that In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant.

But according to Taylor, the conservation of momentum states that if there are no external forces on a pair of particles [in a collection of many particles] then this particle pair's momentum is conserved.

The Taylor's version is stricter than the Wikipedia's since we can't prove newtons third law from Wikipedia's version.

According to Wikipedia $$\sum\vec P=constant \\ \sum\frac{\partial\vec P}{\partial t}=0$$

At this point, both Taylor and Wikipedia agree. But now consider the case of only two particles, we get $$\frac{\partial\vec P_{12}}{\partial t}=-\frac{\partial\vec P_{21}}{\partial t}\tag1$$ Again both agree.

However, if we consider three particles then all that Wikipedia says that $$\frac{\partial\vec P^w_{12}}{\partial t}+\frac{\partial\vec P^w_{21}}{\partial t}+\frac{\partial\vec P^w_{13}}{\partial t}+\frac{\partial\vec P^w_{31}}{\partial t}+\frac{\partial\vec P^w_{23}}{\partial t}+\frac{\partial\vec P^w_{32}}{\partial t}=0$$

Where I put a superscript of “w” to differentiate it from Taylors version, which says that

$$\frac{\partial\vec P^t_{12}}{\partial t}+\frac{\partial\vec P^t_{21}}{\partial t}+\frac{\partial\vec P^t_{13}}{\partial t}+\frac{\partial\vec P^t{31}}{\partial t}+\frac{\partial\vec P^t_{23}}{\partial t}+\frac{\partial\vec P^t_{32}}{\partial t}=0$$

In addition, Taylor assumes that equation 1 still holds, that is $$\frac{\partial\vec P^t_{12}}{\partial t}=-\frac{\partial\vec P^t_{21}}{\partial t} = \frac{\partial\vec P_{12}}{\partial t}=-\frac{\partial\vec P_{21}}{\partial t}$$

Where he mentions that “this is a subtle point: we have assumed switching off other forces have not altered $${\partial\vec P_{12}}/{\partial t}$$and $${\partial\vec P_{21}}/{\partial t}$$. One could imagine a world where this is not true, that is, external forces affected internal forces. However, this seems not to be the case in our world”.

In addition, according to this post it seems there is no justification, however the question remains, what is the exact statement of conservation of momentum?