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?


1 Answer 1


Wikipedia's "in a closed system" and Taylor's "if there are no external forces on a pair of particles" are the same assumption, when applied to a pair of particles. In particular, the assumption is that there are no forces acting on the two particles other than the forces those two particles exert on each other.

Thus when you say "Now, the third law would be proven if we assume that the force between two particles is not altered by the addition of a third particle." -- both wikipedia and Taylor are assuming the statement you made.

  • $\begingroup$ The explicit assumption made by Taylor is that force between 1 and 2 does not change by addition of 3 or equivalently that we can assume that in a system of particles any pair of particles will behave as if it were independent of others, but this is not the same as Wikipedia's definition since there force between two particles may change due to a third one. Please refer to the answer I linked in the post. $\endgroup$ Mar 21 at 11:45
  • $\begingroup$ @GedankenExperimentalist It is the same as wikipedia's definition since if it were the case that "the force between two particles may change due to a third one" then we would not be able to consider the two particles as a closed system. $\endgroup$
    – Andrew
    Mar 21 at 13:26
  • $\begingroup$ I have edited the post to make my question and argument more clear, please refer to it. $\endgroup$ Mar 21 at 16:43

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