The rotating Kepler problem, which arises as a limiting case of the circular restricted three-body problem, is one of the fundamental systems in celestial mechanics. Owing to its complete integrability, its orbits can be computed explicitly.
In this talk, based on my thesis and my previous work published this year, I will describe a classification of the orbits of the rotating Kepler problem in terms of the angular momentum and the Laplace-Runge-Lenz vector. I will then present the computation of the Conley–Zehnder indices of all periodic orbits below the critical energy level. These results reveal, in particular, the relationship between the types of bifurcations occurring in the rotating Kepler problem and the corresponding Conley-Zehnder indices.
If time permits, I would also like to briefly discuss my ongoing work on changes of Conley-Zehnder indices in generic bifurcations, and more broadly on how such bifurcations may be reflected at the level of Floer chain complexes.