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What, specifically, causes newton's third law? For instance, if I push on a wall, why is it that I experience a force in the opposite direction?

I seem to vaguely understand that is has something to do with electronic repulsion or molecular compression (maybe that's completely wrong, I don't know). As a related question, what would happen if two objects that were /infinitely/ immovable (at the molecular level, it cannot be broken or compressed) collided with each-other?

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  • $\begingroup$ A rather big bang would happen I think!:D $\endgroup$ – Harry David Dec 10 '14 at 8:51
  • $\begingroup$ Cause of these forces? Many things. But why Newton's 3rd law??? In a system, the COM cannot be accelerated by internal forces. So, if one force is there, there must be certainly another equal but opposite force on the COM of the system so that it doesn't get accelerated. This is Physics! $\endgroup$ – user36790 Dec 10 '14 at 9:30
  • $\begingroup$ I think my question was worded incorrectly; I am curious specifically about how the forces are transmitted. I suppose it depends on the situation, so I'll refer back to the hand pushing on the wall. $\endgroup$ – Jotak Dec 10 '14 at 22:08
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We start by noting that force is the rate of change of momentum. Let's suppose you and I are floating in space (so we are the only two interacting bodies) and you're pushing me so I feel a force $F_{me}$, then:

$$ F_{me} = \frac{dp_{me}}{dt} $$

where $p_{me}$ is my momentum.

But we know that momentum is conserved, so since you are the only thing interacting with me your momentum, $p_{you}$, must be changing in the opposite sense to balance out the changes in my momentum. In other words:

$$ \frac{dp_{you}}{dt} = - \frac{dp_{me}}{dt} $$

And since force is rate of change of momentum that means there is a force on you:

$$ F_{you} = \frac{dp_{you}}{dt} = - \frac{dp_{me}}{dt} = -F_{me} $$

So the two forces are equal and opposite just as Newton's third law tells us.

The details of exactly how the forces are transmitted depend on exactly how the two bodies are interacting, but whatever the interaction the changes in momentum must be equal and opposite, and therefore the forces are equal and opposite.

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  • $\begingroup$ I have told the same thing but with centre of mass of the system. But I want to know, if conservation of momentum(linear) is always valid, why is Newton's 3rd law not always true?? $\endgroup$ – user36790 Dec 10 '14 at 12:58
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    $\begingroup$ @user36790: When we get down to the quantum regime it starts getting tricky to define what you mean by a force, so classical Newtonian mechanics doesn't apply, or at least not in any simple sense. Apart from such special cases the third law always applies. The question Does Newton's third law apply to momentum or to forces? has more on this. $\endgroup$ – John Rennie Dec 10 '14 at 13:10
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In John's answer he states:

But we know that momentum is conserved, ...

So we can derive Newton's third law from the conservation of momentum.

But you may ask where does momentum conservation come from? Noether's Theorem

If we apply Noether's Theorem with the condition that the Lagrangian is invariant over space then we get that there is a quantity (momentum) which is conserved.

If you construct your Lagrangian such that it varies from place to place you may get a situation which does not obey the Newton's third law. However this is never necessary and rarely desired.

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