Two-center problem with harmonic-like interactions: periodic orbits and non-integrability

TL;DR

This paper analyzes a classical two-center problem with harmonic interactions, demonstrating the existence of bifurcating periodic orbits and proving the system's generic non-integrability through analytical and numerical methods.

Contribution

It provides the first analytical proof of bifurcating periodic orbits and establishes the non-integrability of the system, supported by numerical simulations.

Findings

Existence of bifurcating periodic orbits from equilibrium points

System is generically non-integrable in the Liouville-Arnold sense

Numerical Poincaré sections and Lyapunov exponents confirm analytical results

Abstract

We study the classical planar two-center problem of a particle $m$ subjected to harmonic-like interactions with two fixed centers. For convenient values of the dimensionless parameter of this problem we use the averaging theory for showing analytically the existence of periodic orbits bifurcating from two of the three equilibrium points of the Hamiltonian system modeling this problem. Moreover, it is shown that the system is generically non-integrable in the sense of Liouville-Arnold. The analytical results are complemented by numerical computations of the Poincar\'e sections and Lyapunov exponents. Explicit periodic orbits bifurcating from the equilibrium points are presented as well.