On the isosceles trapezoidal fourbody 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 fourbody 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 fourbody problem (I4BP), where twopoint masses are equal to 1, while the remaining twopoint 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 xaxis. 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 fourbody problem using the surface of section method.
 Publication:

7th International Conference in Astronomy
 Pub Date:
 July 2023
 Bibcode:
 2023aasp.confE..31S
 Keywords:

 trapezoidal fourbody problem;
 quadruple stellar system;
 homogarphic solutions;
 dynamical behavior;
 surface of section