A common first integral from three-body secular theory and Kepler billiards

TL;DR

This work reveals a common first integral $D=C^{2}-2 h A_{1}$ linking the partially-averaged (secular) dynamics of the three-body problem with integrable Kepler billiards, via the projective dynamics of the two-center problem on spaces of constant curvature. By establishing a projective correspondence between Euclidean and curved-space two-center and Lagrange problems, it shows that the spherical energy induces an additional integral for the Euclidean problem and vice versa, with $D$ remaining conserved under averaging. The authors extend these ideas to the Lagrange problem and to hyperbolic spaces, deriving partially-averaged Hamiltonians $E_{Eucl,a}$, $E_{sph,a}$ and their counterparts in $H^{d}$, all admitting $D$ as a first integral. They then construct broad families of integrable billiards on spaces of constant curvature, where walls are drawn from confocal conic families with a Kepler center at a focus, and where $D$ (alongside the Kepler energy) remains conserved. The results unify celestial-mechanical averaging with geometric billiards, providing new integrable systems and deepening the connections between central-force dynamics, projective mappings, and integrable billiards.

Abstract

We observe that a particular first integral of the partially-averaged system in the secular theory of the three-body problem appears also as an important conserved quantity of integrable Kepler billiards. In this note we illustrate their common roots with the projective dynamics of the two-center problem. We then combine these two aspects to define a class of integrable billiard systems on surfaces of constant curvature.

Theorems & Definitions (19)

• Proposition 2.1
• Remark 2.1
• Definition 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.2
• proof
• Definition 2.2