A Perturbed MIC-Kepler Problem and its Reduction to Isoenergetic Orbit Spaces

The hydrogen atom in the presence of an electromagnetic field has been investigated as a kind of a perturbed Kepler problem. The Kepler problem has been generalized to the MIC-Kepler problem, which is a Kepler problem in a monopole field. If perturbation terms are in normal form, both the perturbed Kepler and the perturbed MIC-Kepler problems can be studied on respective isoenergetic orbit spaces after the reduction by the Hamiltonian flow generated by respective unperturbed terms. This thesis aims to analyze a perturbed MIC-Kepler problem on the isoenergetic orbit space in a similar manner to that for the hydrogen atom in the constant electromagnetic field. In this thesis, the perturbed MIC-Kepler problem is investigated in not only negative- but also positive-energy cases. The positive energy case has not been treated in the perturbation of the Kepler problem. To obtain normal form Hamiltonians for the perturbed MIC-Kepler problem, the perturbed MIC-Kepler problem is lifted to a perturbed conformal Kepler problem, which is related to a perturbed harmonic oscillator or a perturbed repulsive one, depending on whether the energy is negative or positive. The BG-normalization can be applied to the perturbed harmonic and repulsive oscillators to put the perturbed Hamiltonians in normal form, and thereby reduced Hamiltonians in normal form are obtained for the perturbed MIC-Kepler problem. The Hamiltonian in normal form for the perturbed MIC-Kepler problem is reduced to that on the isoenergetic orbit space in the negative- and positive-energy cases, which can be described as a dynamical system on a four-dimensional submanifold of $\mathbb{R}^6$ by using the fact that the isoenergetic orbit spaces are realized as co-adjoint orbits of the respective symmetry groups for the MIC-Kepler problem with negative and positive energies. Each of the induced systems on the isoenergetic orbit spaces is approximated to an integrable system after averaging the reduced Hamiltonian.