Conley–Zehnder indices of spatial rotating Kepler problem

In this paper, we study periodic orbits in the spatial rotating Kepler problem from a symplectic-topological perspective. Our first main result provides a complete classification of these orbits via a natural parametrization of the space of Kepler orbits, using angular momentum and the Laplace–Runge–Lenz vector. We then compute the Conley–Zehnder indices of non-degenerate orbits and the Robbin–Salamon indices of degenerate families, establishing their contributions to symplectic homology via the Morse–Bott spectral sequence. To address coordinate degeneracies in the spatial setting, we introduce a new coordinate system based on the Laplace–Runge–Lenz vector. These results offer a full symplectic-topological profile of the three-dimensional rotating Kepler problem and connect it to generators of symplectic homology.