The special concave toric domain for the rotating Kepler problem

TL;DR

This paper constructs a special concave toric domain for the rotating Kepler problem, enabling precise computation of ECH capacities and advancing symplectic embedding analysis in celestial mechanics.

Contribution

It introduces a novel geometric framework and combinatorial tree structure for analyzing symplectic capacities in the RKP at sub-critical energies.

Findings

Construction of a special concave toric domain for RKP

Development of a Stern-Brocot inspired energy encoding tree

Rigorous computation of ECH capacities below critical energy

Abstract

The Rotating Kepler Problem (RKP) arises as a fundamental model in celestial mechanics, appearing as a limiting case of the circular restricted three-body problem. It offers a tractable yet rich framework for studying periodic orbits, energy levels, and symplectic structures. In this work, we investigate the RKP for energy values less than or equal to the critical threshold -3/2. Using the Ligon-Schaaf and Levi-Civita symplectic regularizations, we identify a bounded component of the RKP phase space. Within this setting, we construct a special concave toric domain (SCTD) tailored to the RKP, which provides a concrete geometric framework for computing embedded contact homology (ECH) capacities below the critical energy. The SCTD enables a rigorous analysis of symplectic embedding problems and energy constraints in dynamical systems. Furthermore, we introduce a combinatorial tree structure, inspired by the Stern-Brocot tree, that encodes energy data and facilitates the computation of ECH capacities on the bounded component. These results advance the understanding of symplectic embeddings in celestial mechanics and provide new tools for the study of Hamiltonian dynamics in rotating systems.