Scattering invariants in Euler's two-center problem

TL;DR

This paper investigates the scattering dynamics of Euler's two-center problem in three dimensions, revealing topologically non-trivial scattering monodromy and introducing a general approach for integrable scattering systems.

Contribution

It introduces the concept of scattering monodromy in the spatial two-center problem and presents a general method applicable to integrable scattering systems.

Findings

Identification of scattering monodromy in the spatial two-center problem

Demonstration of topologically non-trivial scattering dynamics

Development of a general approach for integrable scattering systems

Abstract

The problem of two fixed centers was introduced by Euler as early as in 1760. It plays an important role both in celestial mechanics and in the microscopic world. In the present paper we study the spatial problem in the case of arbitrary (both positive and negative) strengths of the centers. Combining techniques from scattering theory and Liouville integrability, we show that this spatial problem has topologically non-trivial scattering dynamics, which we identify as scattering monodromy. The approach that we introduce in this paper applies more generally to scattering systems that are integrable in the Liouville sense.