ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem

TL;DR

This paper investigates the dynamics of the spatial isosceles three-body problem using Embedded Contact Homology, establishing the existence of infinitely many periodic orbits below and above a critical energy level through twist estimates and Reeb flow analysis.

Contribution

It introduces new ECH-based constraints and twist estimates that prove the existence of infinitely many periodic orbits in the spatial isosceles three-body problem.

Findings

Below the critical energy, all compact energy surfaces have infinitely many periodic orbits.

Above the critical energy, there are infinitely many periodic and parabolic trajectories.

The study connects ECH constraints with dynamical properties like twist intervals.

Abstract

We study dynamical constraints arising from Embedded Contact Homology (ECH) in the spatial isosceles three-body problem. For energies below the critical level, the dynamics on the energy surface is identified with a Reeb flow on the tight three-sphere. We obtain quantitative estimates for the Euler orbit, including monotonicity of its transverse rotation number and a strict inequality comparing its action with the contact volume. Combined with the ECH classification of Reeb flows on the tight three-sphere with two simple periodic orbits, these estimates rule out the two-orbit scenario, thus forcing every compact energy surface below the critical level to have infinitely many periodic orbits. The result admits a dynamical interpretation via disk-like global surfaces of section bounded by the Euler orbit. In this setting, the rotation number and the contact volume define a non-trivial twist interval which encodes the relative winding of periodic orbits. For energies above the critical level, where the energy surface is non-compact, we prove the existence of infinitely many periodic orbits and infinitely many parabolic trajectories via twist estimates near infinity.