  The magnetic Aharonov-Bohm (A-B) effect occurs when a point charge interacts
with a line of magnetic flux, while its dual, 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 A-B effects which is
central to both quantum mechanics and electromagnetism may be possible.
