Degeneracy of Planar Central Configurations in the $N$-Body Problem

The degeneracy of central configurations in the planar $N$-body problem makes their enumeration problem hard and the related dynamics appealing. To truly understand the bifurcations of central configurations, we should work in the FULL configuration space which also facilitates the computer-aided methods. The degeneracy is always intertwined with the symmetry of the system of central configurations which makes the problem subtle. By analyzing the Jacobian matrix of the system, we systematically explore the direct method to single out trivial zero eigenvalues associated with translational, rotational and scaling symmetries, thereby isolating the non-trivial part of the Jacobian to study the degeneracy. Four distinct formulations of degeneracy are presented, each tailored to handle different forms of the system appeared in the literature. The method is applied to such well-known examples as Lagrange’s equilateral triangle solutions for arbitrary masses, the square configuration for four equal masses and the equilateral triangle with a central mass revealing specific mass values for which degeneracy occurs. Combining with the interval algorithm, the nondegeneracy of rhombus central configurations for arbitrary mass is also established.

💡 Research Summary

The paper addresses the long‑standing difficulty of determining when a planar central configuration of the Newtonian N‑body problem is degenerate, i.e., when the Jacobian of the defining equations fails to have full rank. Because the central‑configuration equations are invariant under translations, rotations, and (in many formulations) scalings, the Jacobian always possesses a number of trivial zero eigenvalues that obscure the genuine degeneracy. The authors propose a unified, computation‑friendly framework that removes these trivial eigenvalues directly from the Jacobian, thereby isolating the “true” degeneracy.

First, the authors recall the various algebraic forms of the central‑configuration equations that appear in the literature: the Lagrange‑Euler form (force balance with a Lagrange multiplier λ), the constrained form where the moment of inertia I is fixed, and the version with the center of mass set to the origin. Each form admits a different set of symmetries: (i) only rotation, (ii) rotation plus scaling, (iii) translation plus rotation, or (iv) all three symmetries together. Accordingly, the Jacobian may contain one, two, three, or four trivial zero eigenvalues.

The core contribution is a systematic classification into four “forms” of degeneracy:

1. Form I (single trivial zero – rotation only) – By fixing the center of mass and normalising the moment of inertia (I = 1), translation and scaling are eliminated, leaving only rotational invariance. The Jacobian then has exactly one zero eigenvalue; the reduced (2N‑1)×(2N‑1) matrix determines degeneracy.

2. Form II (two trivial zeros – rotation and scaling) – Fixing the center of mass but leaving the scale free removes translation but retains rotation and scaling. Two zero eigenvalues remain; the reduced Jacobian is (2N‑2)×(2N‑2).

3. Form III (three trivial zeros – translation and rotation) – By fixing the scale (I = 1) while keeping the center of mass free, translation and rotation survive, giving three trivial zeros; the reduced Jacobian is (2N‑3)×(2N‑3).

4. Form IV (four trivial zeros – all symmetries) – No constraints are imposed; the full Jacobian has four trivial zeros (two translations, one rotation, one scaling). The reduced Jacobian is (2N‑4)×(2N‑4).

In each case the authors describe how to construct the reduced Jacobian, either by projecting onto the orthogonal complement of the symmetry subspace or by explicitly eliminating variables (e.g., setting two bodies on coordinate axes). Non‑degeneracy is defined as the non‑vanishing of the determinant of the reduced Jacobian; degeneracy occurs when this determinant is zero.

The paper then applies this machinery to several classical configurations:

• Lagrange’s equilateral triangle – For arbitrary positive masses, Form I shows that after removing the single rotational zero, the remaining 5×5 Jacobian has a non‑zero determinant, confirming that the equilateral triangle is always non‑degenerate.

• Square configuration of four equal masses – Using Form II (center‑of‑mass fixed, scaling free) the authors reduce the Jacobian to a 6×6 matrix and compute its determinant analytically; it never vanishes, establishing non‑degeneracy for any equal‑mass square.

• Equilateral triangle with a central mass – This configuration, first studied by Palmore, exhibits a special mass ratio (m_4/m_1 = (2+3\sqrt{3})/(18-5\sqrt{3})) at which an extra zero eigenvalue appears. The authors recover this result in both Form III and Form IV, showing that the configuration is degenerate only at this precise ratio; otherwise it is non‑degenerate.

• Rhombus configurations for four bodies with arbitrary masses – Here the authors combine their reduction (Form IV) with a rigorous interval‑arithmetic algorithm based on the Krawczyk operator. By enclosing all entries of the reduced Jacobian in validated intervals and proving that the interval for its determinant does not contain zero, they obtain a computer‑assisted proof that rhombus central configurations are non‑degenerate for any positive masses.

The interval‑algorithm part is noteworthy: it provides a fully rigorous, computer‑verified proof that avoids the pitfalls of floating‑point rounding. The method systematically subdivides the mass parameter space, computes interval enclosures for the Jacobian entries, and checks that the determinant interval stays away from zero. This approach can be automated and extended to higher‑N problems.

Overall, the paper makes three major contributions:

1. A unified theoretical framework for handling symmetry‑induced trivial eigenvalues in the Jacobian of central‑configuration equations, presented in four clearly distinguished forms.

2. Explicit analytical demonstrations of how the framework resolves degeneracy questions for several classical configurations, recovering known results (e.g., Palmore’s mass ratio) and providing new proofs (e.g., non‑degeneracy of the rhombus for arbitrary masses).

3. A practical, computer‑assisted verification tool based on interval arithmetic that can certify non‑degeneracy in situations where symbolic computation is infeasible.

The authors argue convincingly that studying degeneracy in the full configuration space, rather than in symmetry‑reduced subspaces, is essential because some degeneracies (and the associated bifurcations) are invisible when symmetry constraints are imposed. Their methodology therefore offers a robust pathway for future investigations of central configurations, bifurcation analysis, and the enumeration problem in the planar N‑body problem, and it is readily adaptable to spatial (3‑dimensional) settings and to larger N.