Conley-Zehnder Indices of Spatial Rotating Kepler Problem

TL;DR

This paper provides a comprehensive symplectic-topological analysis of periodic orbits in the spatial rotating Kepler problem, including classification, index calculations, and connections to symplectic homology.

Contribution

It introduces a new coordinate system based on the Laplace-Runge-Lenz vector and fully classifies and computes indices of orbits in the 3D rotating Kepler problem.

Findings

Complete classification of orbits via angular momentum and Laplace-Runge-Lenz vector.

Calculation of Conley-Zehnder and Robbin-Salamon indices for orbits.

Connection of indices to symplectic homology generators.

Abstract

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.