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

Mansur, Abdalla; Shoaib, Muhammad; Szücs-Csillik, Iharka; Offin, Daniel; Brimberg, Jack; Fgaier, Hedia

doi:10.3390/math12193152

Open AccessArticle

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

by

Abdalla Mansur

^1,*

,

Muhammad Shoaib

²

,

Iharka Szücs-Csillik

^3,*

,

Daniel Offin

⁴

,

Jack Brimberg

⁵ and

Hedia Fgaier

⁶

¹

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

²

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

³

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

⁴

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

⁵

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

⁶

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

^*

Authors to whom correspondence should be addressed.

Mathematics 2024, 12(19), 3152; https://doi.org/10.3390/math12193152

Submission received: 3 September 2024 / Revised: 27 September 2024 / Accepted: 5 October 2024 / Published: 9 October 2024

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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.

Keywords: four-body problem; action minimizing orbits; celestial mechanics

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MDPI and ACS Style

Mansur, A.; Shoaib, M.; Szücs-Csillik, I.; Offin, D.; Brimberg, J.; Fgaier, H. Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis. Mathematics 2024, 12, 3152. https://doi.org/10.3390/math12193152

AMA Style

Mansur A, Shoaib M, Szücs-Csillik I, Offin D, Brimberg J, Fgaier H. Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis. Mathematics. 2024; 12(19):3152. https://doi.org/10.3390/math12193152

Chicago/Turabian Style

Mansur, Abdalla, Muhammad Shoaib, Iharka Szücs-Csillik, Daniel Offin, Jack Brimberg, and Hedia Fgaier. 2024. "Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis" Mathematics 12, no. 19: 3152. https://doi.org/10.3390/math12193152

APA Style

Mansur, A., Shoaib, M., Szücs-Csillik, I., Offin, D., Brimberg, J., & Fgaier, H. (2024). Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis. Mathematics, 12(19), 3152. https://doi.org/10.3390/math12193152

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

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