The Conley-Zehnder indices of the rotating Kepler problem

TL;DR

This paper calculates the Conley-Zehnder indices for all periodic orbits in the rotating Kepler problem below a critical energy, revealing the dynamical convexity of the energy hypersurface and identifying a unique orbit with index 3.

Contribution

It provides a complete determination of Conley-Zehnder indices for the rotating Kepler problem's periodic orbits and proves the universal cover's dynamical convexity.

Findings

All periodic orbits' Conley-Zehnder indices are determined.

The universal cover of the energy hypersurface is dynamically convex.

There is exactly one orbit with Conley-Zehnder index 3 in the universal cover.

Abstract

We determine the Conley-Zehnder indices of all periodic orbits of the rotating Kepler problem for energies below the critical Jacobi energy. Consequently, we show the universal cover of the bounded component of the regularized energy hypersurface is dynamically convex. Moreover, in the universal cover there is always precisely one periodic orbit with Conley-Zehnder index 3, namely the lift of the doubly covered retrograde circular orbit.