The Darwin model for space-charge dynamics

Using the method of Green´s functions we found the quasistatic electromagnetic field via four-vector potential for a charge moving along an arbitrary path in a perfectly conducting cylindrical waveguide. Solutions are expressed analytically through the Green´s functions of the Laplace operator with Dirichlet and Neumann boundary conditions. Boundary conditions for a four-vector potential in cylindrical coordinates are obtained via analysis of those for corresponding fields on the perfectly conducting surface and the Coulomb gauge condition. We also propose a method, which enables one to reduce the problem for the vector potential to a system of scalar Laplace equations in cylindrical coordinate system.