The central configurations of four masses x, −x, y, −y

Abstract

The configuration of a homothetic motion in the N -body problem is called a central configuration. In this paper, we prove that there are exactly three planar non-collinear central configurations for masses x, −x, y, −y with x = y (a parallelogram and two trapezoids) and two planar non-collinear central configurations for masses x, −x, x, −x (two diamonds). Except the case studied here, the only known case where the four-body central configurations with non-vanishing masses can be listed is the case with equal masses (A. Albouy, 1995(A. Albouy, -1996, which requires the use of a symbolic computation program. Thanks to a lemma used in the proof of our result, we also show that a co-circular four-body central configuration has nonvanishing total mass or vanishing multiplier.

Symmetry of planar four-body convex central configurations

“…While vorticities may be negative, we always assume m i O0 for all i, 1%i%n. Note that this implies lO0 (for results with some negative masses, see Celli (2007) andO'Neil (2007)). The main result we prove in this article is as follows.…”

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

“…Perez-Chavela and Santoprete [13] show the existence of kite central configurations for a pair of symmetric masses and show that such a configuration must always possess a symmetry. Similarly, Celli [34] proves the existence of planar diamond and trapezoidal central configurations for two pairs of equal masses. Corbera and Llibre [35] give a complete classification of the same problem and show that this setup has exactly 34 different classes of central configurations.…”

Central Configurations of Four Bodies with Gravitational Charges of Both Signs

“…In this paper we assume an equation of motion for the particles different from that used in references [3], [4], [5]. We will show that their equation of motion applied to two particles of opposite charge, leads to a contradiction with Newton's first law of motion: under no external force, a body moves with a constant velocity vector.…”

“…where r j denotes the position vector of particle j in 3D-space, m j is its positive mass, G is the positive constant of universal gravitation, r lj = |r l − r j | is the distance between particles j and l, and e j is the charge of particle j such that m j = |e j |, with two possible choices of sign for the charge. In the mathematical literature we have found some papers in the context of determining Four-Body central configurations that consider negative masses [3], [4], [5], with a different equation of motion, a different definition of central configuration and without distinguishing between masses and charges. The difference with respect to the differential equation of motion used in those papers is stressed in an Appendix at the end of the paper, where, assuming the validity of Newton's second and third laws, the equation of motion used by those Authors is shown to produce, for a system of two masses of equal magnitude but different inertial and gravitational sign, a rigid body which is self-accelerated under no external force, violating Newton's first law expressing that a body subject to no external forces moves with constant velocity.…”