2013/04/23 by Andrzej J. Maciejewski, Maria Przybylska, Leon Simpson +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Celestial mechanics #Classical mechanics #Differential Galois theory #Differential equation #Dumbbell #Embedding problem #Galois group #Geometry #Hamiltonian (control theory) #Hamiltonian system #Massless particle #Mathematical analysis #Mathematical optimization #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Ordinary differential equation #Physics #Point (geometry) #Pure mathematics #Quantum chaos and dynamical systems #math-ph #math.DS #math.MP #msc:12H05 #msc:37J30 #msc:70F07 #msc:70F15 #nlin.CD #nlin.SI
paper · pdf · doi:10.1007/s10569-013-9514-7
published as Celest Mech Dyn Astr (2013) 117:315-330 · 15 pages, 4 figures
arxiv created 2013/04/23 · openalex publication_date 2013/09/02 · arxiv updated 2015/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems.