Colloquium

Normal forms for perturbed MIC-Kepler problems and reduced Hamiltonians on isoenergetic orbit spaces

松本 昇吾

11月26日(金) 13時30分

The MIC-Kepler problem describes motion of a mass around a monopole. The topology of isoenergetic orbit space, which is defined to be a quotient manifold of the energy manifold by the Hamiltonian flow, depends on whether energy is negative, positive, or zero. Flows other than fixed points do not occur on the isoenergetic orbit space, but if the Hamiltonian of the MIC-Kepler problem is perturbed so as to Poisson-commute with itself, the perturbed Hamiltonian reduces to that on the isoenergetic orbit space to give rise to a flow on it. We are interested in such a reduced Hamiltonian in the negative-, positive-, or zero-energy cases.

In this talk, we define a perturbed MIC-Kepler problem to be the MIC-Kepler problem in orthogonal constant magnetic and electric field. Since the perturbed Hamiltonian do not Poisson-commute with the unperturbed Hamiltonian, we calculate the normal form for the perturbed MIC-Kepler problem in order to obtain a Poisson-commuting perturbed Hamiltonian and then to find a reduced Hamiltonian on isoenergetic orbit space.

To calculate the normal form, we introduce the conformal Kepler problem, which can be reduced to the MIC-Kepler problem by an U(1) action. This is because the conformal Kepler problem is related to the 4-dimensional harmonic oscillator, and then calculation of normal form is easy to perform.

We have calculated the normal form for the perturbed MIC-Kepler problem in the negative- and positive-energy cases and have obtained its reduced Hamiltonian defined on isoenergetic orbit spaces. We will show the results which are computed on Maxima.


参考文献:
[1] T. Iwai, A dynamical group SU(2,2) and its use in the MIC-Kepler problem, J. Phys. A: Math. Gen. 26, 609(1993).
[2] Y. Uwano, From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition, J. Phys. A: Math. Gen. 33, 6635(2000).