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Two-center problem with harmonic-like interactions: Periodic orbits and non-integrability.
A M Escobar Ruiz1, Lidia Jiménez-Lara1, J Llibre2
1Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, P.O. Box 55-534, México D.F. 09340, Mexico.
Chaos (Woodbury, N.Y.)
|December 1, 2025
Summary
This study investigates periodic orbits in a classical two-center problem using averaging theory. Analytical and numerical methods reveal bifurcating periodic orbits from equilibrium points in this non-integrable Hamiltonian system.
Area of Science:
- Classical Mechanics
- Dynamical Systems
- Celestial Mechanics
Background:
- The two-center problem is a fundamental model in classical mechanics.
- Understanding periodic orbits is crucial for analyzing complex dynamical systems.
Purpose of the Study:
- To analytically demonstrate the existence of periodic orbits in the classical planar two-center problem.
- To investigate the integrability of the Hamiltonian system modeling this problem.
- To complement analytical findings with numerical simulations.
Main Methods:
- Averaging theory applied to the Hamiltonian system.
- Analytical determination of bifurcating periodic orbits.
- Numerical computation of Poincaré sections and Lyapunov exponents.
Main Results:
- Existence of periodic orbits bifurcating from two of the three equilibrium points is analytically shown.
- The system is demonstrated to be generically non-integrable (Liouville-Arnold sense).
- Explicit periodic orbits are presented through analytical and numerical results.
Conclusions:
- Averaging theory effectively identifies periodic orbits in the two-center problem.
- The system's non-integrability is confirmed, highlighting complex dynamics.
- Combined analytical and numerical approaches provide a comprehensive understanding of the system's behavior.
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