Non-integrability of charged three-body problem

2025/01/30 by Maria Przybylska, Andrzej J. Maciejewski · 1 voice
Engineering · Mathematics · Physics and Astronomy · #Celestial mechanics #Classical mechanics #Mathematical analysis #Mathematics #Nuclear physics research studies #Physics #Quantum chaos and dynamical systems #Spacecraft Dynamics and Control #Three-body problem #Two-body problem #n-body problem #nlin.CD

paper · pdf · open access · doi:10.1007/s10569-025-10237-3

published in Celestial Mechanics and Dynamical Astronomy 137(1) (Springer Science+Business Media)

openalex publication_date 2025/01/30 · arxiv published 2025/04/24 · arxiv updated 2025/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We consider the problem of n points with positive masses interacting pairwise with forces inversely proportional to the distance between them. In particular, it is the classical gravitational, Coulomb or photo-gravitational n-body problem. Under this general form of interaction, we investigate the integrability problem of three bodies. We show that the system is not integrable except in one case when two among three interaction constants vanish. In our investigation, we use the Morales–Ramis theorem concerning the integrability of a natural Hamiltonian system with a homogeneous potential and its generalization.

Citations

• Critical points at infinity in charged N-body systems
• Hill regions of charged three-body systems
• Critical points of the integral map of the charged 3-body problem
• Integrability and non integrability of some n body problems
• Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems
• On a planar circular restricted charged three-body problem
• A note on algebraic potentials and Morales-Ramis theory
• Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom
• Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

Discussions

• Maria Przybylska, Andrzej J. Maciejewski: Non-integrability of charged three-body problem https://arxiv.org/abs/2504.17302 https://arxiv.org/pdf/2504.17302 https://arxiv.org/html/2504.17302 [bsky, 1 points, 1 comments]

Related

• Non-integrability of the n-body problem
• Nonintegrability of the restricted three-body problem
• Integrability and non integrability of some n body problems
• An analytically solvable three-body problem
• On a planar circular restricted charged three-body problem