Central configurations in the trapezoidal four-body problems

Abstract

In this paper we discuss the central configurations of the Trapezoidal four-body Problem. We consider four point masses on the vertices of an isosceles trapezoid with two equal masses m 1 = m 4 at positions (∓0.5, r B ) and m 2 = m 3 at positions (∓α/2, r A ). We derive, both analytically and numerically, regions of central configurations in the phase space where it is possible to choose positive masses. It is also shown that in the compliment of these regions no central configurations are possible.

Central Configurations and Action Minimizing Orbits in Kite Four-Body Problem

“…Since the classification of central configurations as one of the problems for the 21st century by Smale [11], it has attracted a lot of attention in recent years and has helped in the understanding of the n-body problem [12][13][14][15][16][17][18][19][20][21][22][23]. Ji et al [24] and Waldvogel [25] study a rhomboidal four-body problem with two pairs of masses and use Poincaré sections to find regions of stability for the rhomboidal four-body problem.…”

“…e function μ 0 (a) is a bounded, well-defined continuous function of a except when q(a) � a 2 + 1 − ψ(a) 2 � 0. To identify the values of a where q(a) � 0, we write it as q(a) � − 9883.18a 22 e numerical solution of q(a) � 0 shows that it has three real roots a � 1.14605, a � 1.2471, and a � 1.44556. However, a careful observation of the region of existence of central configuration for the four-body problem in Figure 1 shows that a � 1.14605 defines a boundary between the region of existence and nonexistence and a � 1.2471 and a � 1.44556 are outside the domain of interest.…”