The Problem of Two Fixed Centers: Bifurcation Diagram for Positive Energies

TL;DR

This paper provides a detailed analysis of the two fixed centers problem with positive energies, describing the bifurcation diagram and the structure of scattering trajectories in an integrable system.

Contribution

It offers an explicit description of the bifurcation diagram for the Euler-Jacobi problem with arbitrary fixed-center strengths and positive energies, enhancing understanding of its integrable dynamics.

Findings

Explicit bifurcation diagram derived

Structure of scattering trajectories characterized

Analysis applicable to arbitrary fixed-center strengths

Abstract

We give a comprehensive analysis of the Euler-Jacobi problem of motion in the field of two fixed centers with arbitrary relative strength and for positive values of the energy. These systems represent nontrivial examples of integrable dynamics and are analysed from the point of view of the energy-momentum mapping from the phase space to the space of the integration constants. In this setting we describe the structure of the scattering trajectories in phase space and derive an explicit description of the bifurcation diagram, i.e. the set of critical value of the energy-momentum map.