The symmetry of the Kepler problem, the inverse Ligon-Schaaf mapping and the Birkhoff conjecture

TL;DR

This paper explores the inverse Ligon-Schaaf mapping in the Kepler problem, providing a simple derivation, numerical methods for solving the Kepler equation, and evidence related to the Birkhoff conjecture in celestial mechanics.

Contribution

It offers a new, straightforward derivation of the inverse LS mapping and applies numerical techniques to investigate key problems in celestial mechanics.

Findings

Confirmed the significance of the generalized Kepler equation.

Developed numerical methods for orbit calculations and time-of-flight data.

Provided numerical evidence supporting the Birkhoff conjecture.

Abstract

The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex and involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to investigate three applications: 1) solutions of the Kepler function, 2) calculation of orbits including time-of-flight data based on the Delaunay Hamiltonian, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem.