Dynamics of a relativistic charge in the Penning trap

We are interested in the motion of a classical charge within a processing chamber of a Penning trap. We examine the relativistic Lagrangian and Hamiltonian dynamics without any approximations. We show that the radial and axial motions are non-linearly coupled to each other whenever the special relativity is taken into account. As the restoring quadruple potential has the axial symmetry, the dynamics of the system can be reduced to two degrees of freedom. If all the energy of a charge belongs to the axial oscillating mode, its time evolution is described by the nonlinear equation of motion for a simple pendulum. If the whole energy is accumulated in radial oscillating mode, the dynamical system resembles a double pendulum. We demonstrate that the Hamiltonian system is not integrable in the Liouville sense in the class of functions meromorphic in coordinates and momenta. Using Poincaré sections, we show that, in spite of the non-integrability, a large part of the phase space is filled by quasi-periodic solutions that encircle some periodic solutions. We determine numerically characteristic frequencies of these periodic solutions.