Symmetric Central Configurations in the Concave 4-Body Problem with Two Pairs of Equal Masses

TL;DR

The paper addresses the classification of planar concave kite central configurations in the Newtonian 4-body problem with two equal-mass pairs. It shows these configurations form a single-parameter family and employs computer-assisted interval arithmetic and the Krawczyk operator to rigorously classify all such configurations, proving the count is 0, 1, or 2 for any nonnegative mass ratio and revealing a fold bifurcation at $m_0 ≈ 1.0027133$. A complete bifurcation diagram is presented, distinguishing symmetric and asymmetric branches in both reduced and full planar spaces. The work extends prior partial results by providing a precise, fold-type bifurcation point and a full symmetric-space bifurcation picture, advancing the understanding of concave central configurations with two equal-mass pairs.

Abstract

We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of the masses must lie on the base of an isosceles triangle, and the other pair on its symmetric axis with one mass positioned inside the triangle formed by the other three. Using a rigorous computer-assisted analytical approach, we prove that for any non-negative mass ratio, the number of such configurations is either zero, one, or two, which can be viewed as a complete classification of this particular family. Furthermore, we show that the unique configuration corresponding to a specific mass ratio is a fold-type bifurcation point within the reduced subspace. We also give a clear and complete bifurcation picture for both symmetric and asymmetric cases of this concave type in the whole planar 4-body configuration space.

Figures (8)

• Figure 1: A symmetric planar central configuration in the concave 4-body problem with two pairs of equal masses, where $m_1=m_2$ are located on the base and $m_3=m_4$ on its symmetric axis with $m_3$ inside the triangle formed by the other three.
• Figure 3.2: Bisecting $r_{13}$ when $r_{13}=a>2$.
• Figure 3.3: Bisecting $r_{14}$ when $r_{14}=b<2$.
• Figure 3.4: The function $b=\hat{b}(a)$ (the dashed-dotted black curve) and $m=\hat{m}(a)$ (the solid green curve) with $a\in I_A$. The minimum point of $\hat{b}(a)$ is $(a, \hat{b}(a))=$$(1.0068269818055548,1.6641309857549297)$. The maximum point of $\hat{b}(a)$ is $(a, \hat{b}(a))=$$(1.5397067078739939865, 2.4488397355312008965)$. The maximum point of $\hat{m}(a)$ is $(a, \hat{m}(a))=$$(1.1733802447932032924,1.00271332903708271708)$.
• Figure 3.5: Some examples of the concave kite central configurations with two pairs of equal masses. The left and right figures show the positions of masses $m_3$ and $m_4$, respectively. In the left figure, different colors represent different mass values for $m_3$, and the dashed line and the solid line with the same color denote the two distinct solutions, respectively. The right figure uses the same color and line scheme to denote the corresponding positions of $m_4$ for each mass value. For example, the red lines correspond to the unique solution where $m = m_0$. The black dashed and solid lines denote the two distinct solutions for the equal mass case, respectively.

Theorems & Definitions (18)

• Theorem 1
• Remark 1
• Theorem 2
• Lemma 1: Perpendicular Bisector Theorem, moeckel1990
• Lemma 2
• Theorem 3
• Proposition 1
• proof
• Corollary 1
• proof