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

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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 — 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.

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Acknowledgments

The authors would like to thank Isobel Falconer for very useful historical insights and references. Special thanks to Carles Simó, Rodrigo Schaefer, Umberto Hryniewicz, Sergio Joras and Sonia P. de Carvalho for suggestions and encouragements.

Funding

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 Coordenação de Aperfeiçoamento de Pessoal de Ensino Superior (CAPES, Brazil), through the Graduate Program (Programa de Pos-graduação) 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).

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Authors and Affiliations

  1. Departamento de Matemática, Facutad de Ciencias, Universidad del Bii o-Bii o, Casilla 5-C, Concepción, VIII-región, Chile

    Jaime Andrade

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

    Stefanella Boatto & Gladston Duarte

  3. Barcelona Graduate School of Mathematics & Departament de Matemàtiques i Informática, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain

    Stefanella Boatto & Gladston Duarte

  4. Institut de Mathématiques de Bourgogne, Université de Bourgogne, 21078, Dijon, France

    Thierry Combot

  5. Departamento de Fiisica-Matemática, Instituto de Fisica, Universidade Federal de Rio de Janeiro, Rio de Janeiro, RJ, Brazil

    Teresinha J. Stuchi

Authors
  1. Jaime Andrade
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  2. Stefanella Boatto
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  3. Thierry Combot
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  4. Gladston Duarte
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  5. Teresinha J. Stuchi
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Corresponding authors

Correspondence to Jaime Andrade, Stefanella Boatto, Thierry Combot, Gladston Duarte or Teresinha J. Stuchi.

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The authors declare that they have no conflicts of interest.

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Andrade, J., Boatto, S., Combot, T. et al. N-body Dynamics on an Infinite Cylinder: the Topological Signature in the Dynamics. Regul. Chaot. Dyn. 25, 78–110 (2020). https://doi.org/10.1134/S1560354720010086

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  • DOI: https://doi.org/10.1134/S1560354720010086

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