Some special restricted four-body problems—II. From Caledonia to Copenhagen

Collinear Equilibrium Solutions of Four-body Problem

“…If R 1 = R 2 = R 3 , then necessarily R 4 = R 3 = R 2 = R 1 (Roy and Steves 1998). To show that the collinear non-symmetric arrangement of the two pairs of equal masses as shown in Fig.…”

“…In this section we have derived the two symmetric collinear equilibrium solutions of Roy and Steves (1998) and gave a very simple proof of the existence of infinite family of equilibrium solutions.…”

“…Ouyang and Xie (2005) found regions on the configuration space where it is possible to choose masses for collinear configuration of four bodies which will make it central. Roy and Steves (1998) discussed some special analytical solutions of the fourbody problems. They showed that these solutions reduce to the Lagrange solutions of the Copenhagen problem when two of the masses are equally reduced.…”

“…They showed that these solutions reduce to the Lagrange solutions of the Copenhagen problem when two of the masses are equally reduced. In this paper, we will derive the collinear solutions of Roy and Steves (1998). We will discuss the existence of continuous family of equilibrium solutions for the above-mentioned collinear four-body problems which include two symmetric arrangements of two pairs of masses (Section 2) and two non-symmetric arrangement of two pairs of masses (Section 3).…”

Periodic motions of a small body in the Newtonian field of a regular polygonal configuration of ν+1 bodies

“…In the gravitational version of this problem, the system is characterized by two parameters: the mass parameter, which is the ratio of the central mass to a peripheral one (β = m 0 /m) and the number ν of the peripheral primaries. We emphasize the advantage of the regular polygon (or otherwise ring) model which, with the change of its parameters, is reduced to some problems of Celestial Dynamics that appeared in the past in international literature, such as the Copenhagen case of the restricted three-body problem (ν = 2 and β = 0), the restricted four-body problem of Marañhao (1995) (ν = 2, β = 0) and the restricted five-body problem (Ollöngren 1988;Markellos et al 2002) (ν = 3 and β = 0), while two of the basic configurations studied by Roy and Steves (1998) in their Caledonian problem, can also be obtained when ν = 4 and β = 0, or ν = 3 and β = 1. Here, we study the simple periodic orbits of a particle, which moves under the combined gravitational action of 8 much larger bodies (ν = 8) with equal masses that are located at the vertices of a regular octagon, while another body with different mass lies at the center of mass of the system (Fig.…”