Bifurcations of planar balanced configurations for the $n$-body problem in $\mathbb{R}^4$

TL;DR

The paper investigates planar balanced configurations in $\\mathbb{R}^4$ and their bifurcations as the scale parameter $s>1$ varies. By extending a variational- and spectral-flow framework to handle fully degenerate trivial branches, the authors prove the existence of bifurcation points along the planar balanced configuration branch and provide a lower bound on their number, with a robust treatment of Morse-Bott degeneracy. They develop a detailed inertia-index and Hessian analysis for planar configurations, and apply a finite-dimensional bifurcation theory to deduce nontrivial branches bifurcating from planar central configurations. Theoretical results are complemented by numerical simulations for $n=4$ and $n=5$ that illustrate how planar configurations connect to spatial ones (e.g., square to tetrahedron and square pyramid) and reveal stability changes governed by central masses. Additional simulations for collinear configurations in the plane reveal multiple bifurcation points and complex branching, supported by discussions of turning points and potential Conley-index verification.

Abstract

Central configurations play a fundamental role in the Newtonian $n$-body problem, as they give rise to motions in which the configuration evolves while preserving its shape up to rotation and scaling. These include relative equilibria, where the configuration rigidly rotates about the center of mass and each body moves along a circular orbit. For $d\le3$, such motions originate only from planar central configurations, whereas in higher dimensions the richer structure of the orthogonal group admits new balanced configurations that can produce non-planar relative equilibria. Building on the framework introduced by Asselle, Portaluri and Fenucci [J. Fixed Point Theory App., 2022], we analyze bifurcations of planar balanced configurations in $\mathbb{R}^4$. We extend a classical variational result, which guarantees the existence of bifurcation points along trivial branches of critical points that are degenerate only at finitely many points, to the case where the trivial branch remains degenerate throughout. Applying this extension, we establish the existence of bifurcation points along the planar balanced configuration branch and derive a lower bound on their number.

Figures (9)

• Figure 1: Non-trivial branch of balanced configurations bifurcating from the square configuration with four equal masses. As $s$ decreases from $\bar{s}=1.4$, the square deforms into a tetrahedron, which becomes regular at $s=1$. All configurations along this branch are minima of $U|_\mathcal{S}$.
• Figure 2: Non-trivial branch bifurcating from the triangular configuration with a central mass. The branch originates at $\bar{s}=2.5$. For equal masses, all solutions along the branch are saddles. Varying the central mass reveals a critical value $m^*$ at which the stability type changes.
• Figure 3: Non-trivial branch for five equal masses originating at $s_1 = 2.5$. As $s$ decreases, the configuration evolves from a square with a central mass into a regular square pyramid at $s=1$. These solutions are minima of $U|_\mathcal{S}$.
• Figure 4: Non-trivial branch for five equal masses originating at $s_2 = 1.2$. As $s$ decreases, the configuration becomes a tetrahedron with the fifth mass at the barycenter. These solutions are saddles of $U|_\mathcal{S}$.
• Figure 5: Non-trivial branches for four equal masses. The branch from $s_1=4.15$ passes through a rhombus configuration before returning to a collinear alignment, with the two central masses exchanged; all solutions are minima. The branch from $s_2=2.4$ also returns to a collinear alignment but with both pairs swapped; all solutions are saddles.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Remark 3.1
• Remark 3.2
• Proposition 3.3
• proof
• Definition 4.1