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I was reading Goldstein's Classical Mechanics vol.2 to brush up, and didn't get far before I got stuck. The book warns that both the weak an strong forms of the action/reaction principle can be broken when forces predicted by the Biot-Savart law are involved. The offending passage is partially copied below (page 6).

If two charges are moving uniformly with parallel velocity vectors that are not perpendicular to the line joining the charges, then the mutual forces are equal and opposite but do not lie along the vector between the charges.[\quote]

So, I draw this system, with Particle 1 at the origin, and Particle 2 at x=y=1. They are both moving in the positive x-direction. Now, they will be repelled from each other due to Coulomb forces, but these forces point along the line between the charges so there is no problem. But since they are traveling in the x-direction, each will generate a magnetic field according to the right-hand rule, spiraling around the x-axis and the y=1 line. They will be influenced by each other's field and experience Lorentz forces so that P1 sees a force pointing upward, and P2 sees a force pointing downward. These forces are equal and opposite, but do not travel through the line between the charges, thus violating the weak principle of action and reaction. So, I see what he did there, but I have two issues.

Issue 1: The particles at the beginning of the problem are in uniform motion. However, since they are experiencing the Lorentz forces, they are accelerating, in particular, the angular momentum of the system is changing since the Lorentz forces form a torque. Angular momentum is conserved. If the angular momentum is changing, where is it coming from?

Issue 2: Suppose I transform to a coordinate system that is moving with the particles so that P1 is at the origin and P2 is not just sitting at x=y=1. The particles will be repelled from each other by the Coulomb force, but where is the magnetic force that was present in the first coordinate system? I should be able to transform coordinates systems and not have forces appear and disappear, they must manifest themselves as something else like a fictitious force or something, but I don't see it when I transform to moving coordinate system. The magnetic forces of the moving charges and thus the Lorentz forces of them on each other seems to just disappear.

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# Newton's third breakage in Goldstein's classical mechanics

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