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Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis

Author

Listed:
  • Abdalla Mansur

    (Libyan Center for Engineering Research and Information Technology, Bani Waleed 637211, Libya)

  • Muhammad Shoaib

    (Smart and Scientific Solutions, 32 Allerdyce Drive, Glasgow G15 6RY, Scotland, UK)

  • Iharka Szücs-Csillik

    (Romanian Academy, Astronomical Institute, Astronomical Observatory Cluj-Napoca, 400487 Cluj-Napoca, Romania)

  • Daniel Offin

    (Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada)

  • Jack Brimberg

    (Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada)

  • Hedia Fgaier

    (Department of Mathematics, Full Sail University, Winter Park, FL 32792, USA)

Abstract

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems.

Suggested Citation

  • Abdalla Mansur & Muhammad Shoaib & Iharka Szücs-Csillik & Daniel Offin & Jack Brimberg & Hedia Fgaier, 2024. "Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis," Mathematics, MDPI, vol. 12(19), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3152-:d:1494550
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