Poisson mechanics for perturbed MIC-Kepler problems at both positive and negative energies

The MIC-Kepler problem, an extension of the Kepler problem, is known to admit the symmetry group SU(2) x SU(2) or SL(2, C), according to whether the energy is negative or positive. In general, each of the co-adjoint orbits of a Lie group carries the canonical symplectic form called the KKS form, and a Hamiltonian dynamical system is defined on it if a suitable Hamiltonian is given. Perturbed MIC-Kepler problems can be treated in this setting if a perturbed Hamiltonian in normal form is determined according to whether the energy is negative or positive. Since the co-adjoint orbit in question can be viewed as a symplectic leaf of the associated Lie algebra su(2) circle plus su(2) or sl(2, C) according to whether the energy is negative or positive, the perturbed MIC-Kepler problems in normal form can be described in the Poisson mechanics defined on respective symmetry Lie algebras. Thus, the equations of motion for perturbed systems can be described in the form of Poisson brackets for both cases of su(2) circle plus su(2) and sl(2, C) on an equal footing. It will be shown further how two parameters assigning a co-adjoint orbit of SU(2) x SU(2) or SL(2, C) are related to the parameters contained in the MIC-Kepler problem. The perturbation of the MIC-Kepler problem to be treated in this article is rather restricted to that caused by the presence of weak constant electric and magnetic fields orthogonal to each other. When regularized, the perturbed Hamiltonians at both positive and negative energies are put in the Birkhoff-Gustavson normal form and thereby the flows generated by the perturbed Hamiltonians are studied in Poisson mechanics in terms of variables associated with constants of motion for the MIC-Kepler problem.