Proving Newton's 3rd law from conservation of momentum

[Quantum55151]

Homework Statement

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 F12 = - F21.]

Relevant Equations

dP/dt = 0 iff Fext = 0
F12 = - F21

I don't quite understand the "subtle point" at the end of the author's solution. Ok, let's imagine for a second that the external forces have an impact on the internal forces. How does that change the mathematical result that the two forces are equal and opposite to each other? Even if, hypothetically, we lived in a world where "the presence or absence of external forces affected internal forces", the magnitude or direction of the forces could potentially change change, but the relation between the internal forces, i.e. Newton's 3rd law, would still hold...

Or am I missing something?

 

[Lnewqban]

The way I see it:
If an external force is adding momentum to only one of the particles, those contact equal forces would lose the previous balance, and the two particles would tend to separate from each other.

I imagine one perfectly isolated iron particle in contact with one carbon particle, attracting each other by their own gravitational effect.
F iron on carbon = F carbon on iron

If a magnetic field is allowed to reach both previously isolate particles, then,
Fnet iron on carbon < F carbon on iron

 

[haruspex]

Even if, hypothetically, we lived in a world where "the presence or absence of external forces affected internal forces", the magnitude or direction of the forces could potentially change change, but the relation between the internal forces, i.e. Newton's 3rd law, would still hold

Possibly, but the author's point is that the proof given depends on the assumption that those other forces can be switched off without affecting the two in question.
E.g. consider three particles where F12=-F21+x, F23=-F32+y, F31=-F13-x-y. Momentum is conserved. But if, by some magic, if you take any one particle away then the forces between the remaining two become equal and opposite, so momentum is also conserved for each subsystem.