On the isosceles trapezoidal four-body problem
Abstract
The orbits of stars conserve information about the formation processes of the multiple star systems and exploring dynamically their motion helps us to understand the evolution of stars. Accordingly, special types of the four-body problem investigated analytically and numerically can provide a better understanding of the dynamical behaviour of quadruple stellar systems. This paper deals with the isosceles trapezoidal four-body problem (I4BP), where two-point masses are equal to 1, while the remaining two-point masses are equal to m. Consider these four point masses on the vertices of a trapezoid. Let us take the distance between the first two bodies to be 1 and between the last two bodies equal to a, and the first two particles lying on the x-axis. The length of each side is b. The position of the bodies can be marked as follows r_1(‑1/2, 0), r_2(1/2, 0), r_3(a/2, (b^2‑(1‑a)^2/4)^1/2), and r_4(-a/2, (b^2‑(1‑a)^2/4)^1/2). Hence, the time evolution of the system is uniquely defined by the geometrically reduced Hamilton's equations. Further, we studied the minimizing property of the solutions and demonstrated that the minimizers of the action functional restricted to homographic solutions are the Keplerian elliptical solutions, and this functional has a minimum. In addition, we investigated the dynamical behaviour of the isosceles trapezoidal four-body problem using the surface of section method.
- Publication:
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7th International Conference in Astronomy
- Pub Date:
- July 2023
- Bibcode:
- 2023aasp.confE..31S
- Keywords:
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- trapezoidal four-body problem;
- quadruple stellar system;
- homogarphic solutions;
- dynamical behavior;
- surface of section