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Let's say I have two particles which exert a mysterious force between them. Say they lie on the x-axis. Newton's 3rd law tells us that the force on the two particles due to the other must be opposite to each other, for example they are both attracted or repelled (F on the x-axis), but why can't the force have a component in the y-direction, creating a net torque?

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    $\begingroup$ Does "particle" here mean a point particle only? $\endgroup$ – BowlOfRed Jan 23 '16 at 0:44
  • $\begingroup$ If your "particles" have a dipole moment, it's absolutely possible for them to exert a torque... $\endgroup$ – Floris Jan 23 '16 at 2:18
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Ultimately, the third law expresses in Newton's formalism the principle of conservation of momentum.

In advanced physics you will understand that the laws of physics of the world can be defined in terms of "Lagrangians" and "action principles". An action principle takes a bunch of trajectories of particles from some starting positions to their ending positions (a "path" for the "system of particles" through their "phase space" $\{x_i, v_i\}$) and assigns that path a number, called the action of the path.

The law of stationary action, which subsumes all other laws of physics, then says: of all the paths that the system could take through phase space, the one that it does take has the special property that "nearby" paths all have the same actions. Often this is called the principle of "least" action since the easiest way to do this is for the action to be minimized.

Once you have this mathematical formalism, any particular laws of physics can be entirely reduced to figuring out these action principles. But there is something much more powerful which happens, called Noether's theorem, named after the mathematician Emmy Noether who discovered it. This theorem says: any continuous symmetry in your action principle generates a conserved quantity. You can also use it backwards and look for what symmetries generate known conserved quantities.

Conservation of momentum turns out to correspond to the symmetry "the laws of physics [i.e. the action principle] are the same under translations in space." So the third law ultimately expresses "the laws of physics are the same here as they are there."

Conservation of energy turns out to be the symmetry under translation in time: the laws of physics are the same from moment to moment. And conservation of angular momentum is the symmetry under rotations of space.

You probably have a long way to go before you are dealing with action principles, but that is the deepest reason that we know of why momentum is conserved, and therefore why every force comes in a logical pair with an equal and opposite force.

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Symmetry, no direction is in any way special so the particles cannot decide which direction the torque can it be, spontaneous symmetry breaking is impossible because this torque apparently is large before anything moves.

Also, this will violate conservation of angular momentum because the angular momentum around the centre of mass will change (if one mass moves in +x-direction, the other moves in -x direction (third law) and angular momentum vector in z-axis appears).

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