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N -body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
Regular and Chaotic Dynamics ( IF 1.285 ) Pub Date : 2020-02-20 , DOI: 10.1134/s1560354720010086
Jaime Andrade, Stefanella Boatto, Thierry Combot, Gladston Duarte, Teresinha J. Stuchi

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.
更新日期:2020-02-20

 

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