[Paper Review] Molecules of the Euler problem of two fixed centers and its applications

Seongchan Kim|arXiv (Cornell University)|Jun 17, 2016

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper investigates negative energy hypersurfaces in the Euler problem of two fixed centers, using elliptic coordinates to analyze periodic and homoclinic orbits. It proves that all doubly-covered periodic orbits below the critical Jacobi energy are torus knots and demonstrates that homoclinic orbits exist for all mass ratios and Lyapunov orbit energies, with stable and unstable manifolds coinciding and each orbit rotating exactly once around one primary.

ABSTRACT

We study the molecules of negative energy hypersurfaces of the Euler problem. As an application, we determine the knot types of periodic orbits: we show that for energies below the critical Jacobi energy, every doubly-covered periodic orbit in the regularized system is a torus knot. Moreover, we prove that in the Euler problem homoclinic orbits exist for all mass ratios and for all energies at which the Lyapunov orbit exists. In particular, the unstable and the stable manifolds of the Lyapunov orbit coincide with each other. Moreover, by means of the elliptic coordinates, we show that every homoclinic orbit rotates around one of the primaries precisely once.

Motivation & Objective

• To classify the knot types of periodic orbits in the regularized Euler problem of two fixed centers.
• To establish the existence and geometric structure of homoclinic orbits across all mass ratios and Lyapunov orbit energies.
• To determine the rotational behavior of homoclinic orbits around the primaries using elliptic coordinates.
• To characterize the topological and dynamical properties of negative energy hypersurfaces in the two-center problem.

Proposed method

• The analysis is conducted on negative energy hypersurfaces of the Euler problem using elliptic coordinates to simplify the dynamical system.
• The study employs regularization techniques to handle singularities and analyze doubly-covered periodic orbits.
• Topological classification of periodic orbits is achieved by examining their embedding in the phase space via knot theory.
• Homoclinic orbits are analyzed by studying the intersection of stable and unstable manifolds of the Lyapunov orbit.
• The rotational behavior of homoclinic orbits is determined by tracking their winding number around one of the primaries in elliptic coordinates.
• The critical Jacobi energy is used as a threshold to distinguish between different dynamical regimes in the system.

Experimental results

Research questions

• RQ1 What knot type do doubly-covered periodic orbits in the regularized Euler problem have when energy is below the critical Jacobi energy?
• RQ2 For which mass ratios and energies do homoclinic orbits exist in the Euler problem of two fixed centers?
• RQ3 Under what conditions do the stable and unstable manifolds of the Lyapunov orbit coincide?
• RQ4 How many times does each homoclinic orbit rotate around one of the primaries?
• RQ5 What role do elliptic coordinates play in characterizing the global structure of homoclinic orbits?

Key findings

• All doubly-covered periodic orbits in the regularized system with energy below the critical Jacobi energy are torus knots.
• Homoclinic orbits exist for all mass ratios and for all energies at which the Lyapunov orbit exists.
• The stable and unstable manifolds of the Lyapunov orbit coincide, indicating a homoclinic connection.
• Each homoclinic orbit rotates exactly once around one of the primaries.
• The use of elliptic coordinates confirms that every homoclinic orbit completes a single full rotation around a primary before returning.
• The topological structure of the system is fully characterized by the interplay between energy levels, mass ratios, and orbital winding.