Аннотация:
The formulation of the dynamics of N-bodies on the surface of an infinite cylinder is considered. We have chosen such a surface to be able to study the impact of the surface’s topology in the particle’s dynamics. For this purpose we need to make a choice of how to generalize the notion of gravitational potential on a general manifold. Following Boatto, Dritschel and Schaefer [5], we define a gravitational potential as an attractive central force which obeys Maxwell’s like formulas.
As a result of our theoretical differential Galois theory and numerical study — Poincaré sections, we prove that the two-body dynamics is not integrable. Moreover, for very low energies, when the bodies are restricted to a small region, the topological signature of the cylinder is still present in the dynamics. A perturbative expansion is derived for the force between the two bodies. Such a force can be viewed as the planar limit plus the topological perturbation. Finally, a polygonal configuration of identical masses (identical charges or identical vortices) is proved to be an unstable relative equilibrium for all N>2.
Ключевые слова:N-body problem, Hodge decomposition, central forces on manifolds, topology and integrability, differential Galois theory, Poincaré sections, stability of relative equilibria.
Jaime Andrade was partially supported by CONICYT (Chile) through FONDECYT project
11180776. Stefanella Boatto was partially supported by the Luís Santaló Visiting Professor
fellowship through CRM (Catalonia, Spain). Gladston Duarte was partially supported by a
scholarship from the Coordenacão de Aperfeicoamento de Pessoal de Ensino Superior (CAPES,
Brazil), through the Graduate Program (Programa de Pos-graduacão) of the Mathematical Institute
of the Federal University of Rio de Janeiro, and by the María de Maeztu Unit of Excellence
in Research Program (MTM-2014-0445) through the Barcelona Graduate School of Mathematics
(BGSMath).
Поступила в редакцию: 24.12.2019 Принята в печать: 10.01.2020
\RBibitem{AndBoaCom20}
\by Jaime Andrade, Stefanella Boatto, Thierry Combot, Gladston Duarte, Teresinha J. Stuchi
\paper $N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
\jour Regul. Chaotic Dyn.
\yr 2020
\vol 25
\issue 1
\pages 78--110
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\crossref{https://doi.org/10.1134/S1560354720010086}
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Эта публикация цитируется в следующих 1 статьяx:
Jaime Andrade, Stefanella Boatto, F. Crespo, D.E. Espejo, “On the stability of ring relative equilibria in the N-body problem on with Hodge potential”, Can. J. Math.-J. Can. Math., 2023, 1
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