Dynamical convexity of the Euler problem of two fixed centers

TL;DR

This paper analyzes the rotation functions and bifurcations of periodic orbits in the Euler problem of two fixed centers, providing explicit formulas for Conley-Zehnder indices and proving dynamical convexity below a critical energy.

Contribution

It offers a detailed analysis of the rotation functions, explicit formulas for Conley-Zehnder indices, and establishes dynamical convexity of the energy hypersurface in the Euler problem.

Findings

Explicit formulas for Conley-Zehnder indices of collision orbits

Dynamical convexity of the energy hypersurface below critical energy

Analysis of bifurcations of periodic orbits

Abstract

We give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley-Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularized energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.