Relativistic Lagrangian and Hamiltonian Description of a Beam with Space-Charge

Application of the symplectic particle-in-cell algorithm proposed by H.~Qin {\it et al.} to space-charge in cyclotrons.
Comment: Compared to previous version: added one reference, added missing constants (vacuum permittivity and permeability) in equations, added a short conclusion section, rephrased bullet 4 in introduction, rephrased the end of section 2.3, corrected typos