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Gladston Duarte

68530, Rio de Janeiro, RJ, Brazil
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal de Rio de Janeiro


Andrade J., Boatto S., Combot T., Duarte G., Stuchi T. J.
$N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics
2020, vol. 25, no. 1, pp.  78-110
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 $N >2$.
Keywords: $N$-body problem, Hodge decomposition, central forces on manifolds, topology and integrability, differential Galois theory, Poincaré sections, stability of relative equilibria
Citation: Andrade J., Boatto S., Combot T., Duarte G., Stuchi T. J.,  $N$-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics, Regular and Chaotic Dynamics, 2020, vol. 25, no. 1, pp. 78-110

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