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I am reading "Classical Mechanics" by John Taylor.

There is the following problem in this book:

In Section 1.5 we proved that Newton's third law implies the conservation of momentum. Prove the converse, that if the law of conservation of momentum applies to every possible group of particles, then the interparticle forces must obey the third law. [Hint: However many particles your system contains, you can focus your attention on just two of them. (Call them $1$ and $2$.) The law of conservation of momentum says that if there are no external forces on this pair of particles, then their total momentum must be constant. Use this to prove that $F_{12} = -F_{21}$.]

It is true that $F_{12} = -F_{21}$ holds when there are no external forces on this pair of particles.

Then, can we say the following statement?

$F_{12} = -F_{21}$ holds when there exist external forces on this pair of particles.

tchappy ha
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1 Answers1

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By notation external forces are not included, so $F_{21} + F_{12} = 0$ always holds, even if there are external forces. It is $F_{1} + F_{2}$ that is not zero if there is an external force that is not opposite and equal on both particles. Note that the magnetic part of the Lorentz "force" does not obey the third law and does not conserve momentum.

my2cts
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