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For the rate of change of angular momentum of a system to be equal to the torque due to external forces, the torque due to internal forces should be zero. This will mathematically be possible only when the internal forces act along the line joining the particles.

Is this condition valid in all the cases? If not kindly explain with examples.

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The forces between pairs of magnetic dipoles are not central, and in general neither are the chemical bonding forces that hold things together.

That central forces imply that angular momentum is conserved is true. The converse, that angular momentum conservation implies central forces, is not true.

What leads to angular momentum conservation is that that spatial rotations are a symmetry of the laws of nature, and total angular momentum is the associated conserved Noether charge. In other words rotational symmetry implies a "conspiracy" between all the different types of forces so that in an isolated system their net torque is zero.

As an example of such a "conspiracy" consider two bar magnets glued to opposite ends of a ruler. If the magnets are not alligned antiparallel, the forces that each magnet exerts on the other are not central and so they provide a non-zero moment about the centre of the ruler. However the magnets also experience torques that try to make them align antiparallel. Via the glue, these torques are transmitted to the ruler and the total rotational moments about the centre (and indeed about any point) add to zero.

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  • $\begingroup$ Thanks for the examples, can you confirm whether torque due to internal forces is always zero? $\endgroup$ – sheshin Sep 24 '19 at 3:23
  • $\begingroup$ I added a bit more to my answer to give an example. $\endgroup$ – mike stone Sep 24 '19 at 13:38
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It is a useful simplifying assumption in mechanics where it has a broad applicability, but it is not generally true in physics, for the reasons of finite speed of interaction.

In EM theory, finite speed of interaction means that when particle 1 makes a kink in its trajectory due to local external force, this change of motion cannot have any instantaneous influence on the motion of the particle 2 and thus by necessity, cannot have any instantaneous influence on the force $\mathbf F_{1-2}$ due to particle 1 acting on particle 2. But the line joining the two particles is instantaneously influenced. Thus the forces cannot always be central.

The prime example is magnetic force between charged particles, which is always perpendicular to velocity of the subject particle. For example, if two particles orbit each other in circles, the magnetic forces lie in plane perpendicular to the joining line between the particles.

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