Let us consider two point charges, one positive, one negative, interacting via Coulomb force.
In the absence of any other force, this system constitutes an elementary example of two-body problem, and it is therefore integrable.
Let us add a perfectly conducting shield, e.g. an unbounded planar surface. Now, according to the well-known method of virtual charges, each of the two physical charges "sees" also its image charge and the image of the other charge.
Effectively, upon the introduction of the conducting shield, there are four charges, although the position of two of them is constrained, i.e. it just depends on the position of the two real charges. On the other hand, the presence of the mirror introduces other forces on the two physical charges which make the scenario more complex than the basic two-vortex problem.
My question is: is this two-body problem + shield still integrable? Or is it more similar to a three-body problem and therefore chaotic?
