Action minimizing orbits in the trapezoidal four body problem

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

doi:10.3934/math.2023901

AIMS Mathematics

2023, Volume 8, Issue 8: 17650-17665. doi: 10.3934/math.2023901

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Research article

Action minimizing orbits in the trapezoidal four body problem

1.
College of Science, Bani Waleed University, Bani Waleed, Libya
2.
Engineering and Information Technology Research Center, Bani Waleed, Libya
3.
Smart and Scientific Solutions, 32 Allerdyce Drive, Glasgow, G15 6RY, United Kingdom
4.
Romanian Academy, Astronomical Institute, Cluj-Napoca, Romania
5.
Queen's University, Mathematics and Statistics department, Kingston, Ontario, Canada
6.
The Royal Military College of Canada, Mathematics and Computer Science department, Kingston, Ontario, Canada

Received: 27 March 2023 Revised: 03 May 2023 Accepted: 06 May 2023 Published: 23 May 2023
MSC : 34C27, 34C35, 54H20

In this paper, we study the minimizing property for the isosceles trapezoid solutions of the four-body problem. We prove that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum equal to $ \frac{3}{2}(2\pi)^{2/3}T^{1/3}\left(\frac{\xi (a, b)}{\eta (a, b)}\right) ^{2/3} $. Further, we investigate the dynamical behavior in the trapezoidal four-body problem using the Poincaré surface of section method.
- central configuration,
- four-body problem,
- minimizers,
- trapezoidal problem,
- action functional,
- homographic solutions
Citation: Abdalla Mansur, Muhammad Shoaib, Iharka Szücs-Csillik, Daniel Offin, Jack Brimberg. Action minimizing orbits in the trapezoidal four body problem[J]. AIMS Mathematics, 2023, 8(8): 17650-17665. doi: 10.3934/math.2023901

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Abstract

In this paper, we study the minimizing property for the isosceles trapezoid solutions of the four-body problem. We prove that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum equal to $ \frac{3}{2}(2\pi)^{2/3}T^{1/3}\left(\frac{\xi (a, b)}{\eta (a, b)}\right) ^{2/3} $. Further, we investigate the dynamical behavior in the trapezoidal four-body problem using the Poincaré surface of section method.

References

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