The magnetic Aharonov-Bohm (A-B) effect occurs when a point charge interacts with a line of magnetic flux, while its reciprocal, the Aharonov-Casher (A-C) effect, occurs when a magnetic moment interacts with a line of charge. For the two interacting parts of these physical systems, the equations of motion are discussed in this paper. The generally accepted claim is that both parts of these systems do not accelerate, while Boyer has claimed that both parts of these systems do accelerate. Using the Euler-Lagrange equations we predict that in the case of unconstrained motion, only one part of each system accelerates, while momentum remains conserved. This prediction requires a time-dependent electromagnetic momentum. For our analysis of unconstrained motion, the A-B effects are then examples of the Feynman paradox. In the case of constrained motion, the Euler-Lagrange equations give no forces, in agreement with the generally accepted analysis. The quantum mechanical A-B and A-C phase shifts are independent of the treatment of constraint. Nevertheless, experimental testing of the above ideas and further understanding of the A-B effects that are central to both quantum mechanics and electromagnetism could be possible.

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