Pair Space in Classical Mechanics III. Some Four-Body Central Configurations

TL;DR

The paper extends the pair-space approach to the four-body problem, deriving CC conditions as $\ddot{\bm{q}}_{ij}=-\lambda \bm{q}_{ij}$ and leveraging mass-independent relations to classify configurations. It proves that non-planar CCs must be tetrahedra, while planar CCs include equilateral-triangle, isosceles-kite, and isosceles trapezium shapes, with precise mass–shape relations expressed in terms of angles $\alpha$ and $\beta$. The results yield an exhaustive, geometry-driven taxonomy: tetrahedron, equilateral-triangle with a central mass, convex/concave kites, and isosceles trapezium, with parallelogram collapsing to a rhombus; mass ratios determine shape within each class. This pair-space classification provides a systematic framework for identifying all four-body central configurations under the constraint of multiple equal pair distances and sets the stage for exploring configurations with all pair distances distinct.

Abstract

We study central configurations in the four body problem, i.e., configurations in which the forces on all the bodies point to a fixed, single point in space. The newly formulated pair-space formalism yields a set of vectorial equations that fully characterize such configurations. We investigate a sub-class of solutions in which at least two pairs of inter-body distances are equal. The only such non-collinear configurations are the tetrahedron (the unique non-planar configuration), kites and the isosceles trapezium. The specific shapes (internal angles) are determined by the ratio of the masses of the bodies. Mathematical expression are given for all these relations.

Figures (12)

• Figure 1: The bodies $m_1 , m_2 , m_3$ are the vertices of an equilateral triangle inscribed in a circle centered on $m_4$
• Figure 2: The first of three possible configurations of a kite. $m_4$ is outside the triangle $\triangle 123$, to its left, and the kite is convex. The angle $\alpha$, between $\bm{q}_{12}$ and $\bm{q}_{23}$ is marked with a double line, while the angle $\beta$, between $\bm{q}_{24}$ and $\bm{q}_{23}$, is marked with a single line.
• Figure 3: The areas of possible angles for a central kite configuration in the case of a convex kite. The allowed areas are shaded in the graph. The lines bordering them are outside the allowed range as they correspond to singular conditions, such as some masses vanishing or two masses sitting one on top of the other. We explicitly exclude such configurations here.
• Figure 4: The rhombus configuration. Up to scaling, the figure is entirely determined by the angle $\alpha$.
• Figure 5: The second possible kite configuration. $m_4$ and $m_1$ are on the same side of $\bm{q}_{12}$ and $m_4$ is inside the triangle. The resulting kite is concave. As before, the angle $\alpha$, between $\bm{q}_{12}$ and $\bm{q}_{23}$, is marked with a double line, while the angle $\beta$, between $\bm{q}_{24}$ and $\bm{q}_{23}$, is marked with a single line. Note that in this case $\beta < \alpha$.