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N-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics

Abstract : 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 - Poincare 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.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02521855
Contributeur : Imb - Université de Bourgogne <>
Soumis le : vendredi 27 mars 2020 - 16:23:19
Dernière modification le : samedi 28 mars 2020 - 01:57:22

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Jaime Andrade, Stefanella Boatto, Thierry Combot, Gladston Duarte, Teresinha Stuchi. N-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics. Regular and Chaotic Dynamics, MAIK Nauka/Interperiodica, 2020, 25 (1), pp.78-110. ⟨10.1134/S1560354720010086⟩. ⟨hal-02521855⟩

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