Relative equilibria, linear stability and electromagnetic curvature

Giorgia Testolina; Luca Asselle

arxiv: 2604.07975 · v1 · submitted 2026-04-09 · 🧮 math.DS

Relative equilibria, linear stability and electromagnetic curvature

Luca Asselle , Giorgia Testolina This is my paper

Pith reviewed 2026-05-10 17:59 UTC · model grok-4.3

classification 🧮 math.DS

keywords relative equilibrialinear stabilityn-body problemelectromagnetic curvatureRouth criterionMañé critical valuesymplectic planescentral configurations

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The pith

Electromagnetic curvature detects linear stability thresholds for relative equilibria in the n-body problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a geometric criterion for linear stability of relative equilibria by viewing the Newtonian n-body problem through an electromagnetic systems lens. In a two-dimensional model, stability relates to the Mañé critical value and to topological changes in the zero set of the electromagnetic curvature. This recovers Routh's classical criterion for the planar three-body problem and supplies an instability criterion whenever the reduced linearized dynamics splits along invariant symplectic planes. A reader would care because the approach supplies a fresh geometric handle on stability questions, including those connected to Moeckel's conjecture on central configurations.

Core claim

The authors model relative equilibria in the n-body problem as electromagnetic systems. They first study how ambient dimension affects stability by considering the Lagrange equilateral triangle solutions of the three-body problem in four-dimensional space. In the two-dimensional model they connect linear stability both to the Mañé critical value and to the topology of the zero set of electromagnetic curvature, noting that this zero set changes topology precisely at the stability threshold. When applied to the planar n-body problem the same criterion recovers Routh's result for three bodies and yields an instability test for any relative equilibrium whose reduced linearized dynamics splits on

What carries the argument

The electromagnetic curvature, whose zero-set topology changes at the linear stability threshold of relative equilibria.

If this is right

Routh's classical stability criterion for the planar three-body problem follows directly from the electromagnetic curvature test.
Any relative equilibrium whose reduced linearized dynamics splits along invariant symplectic planes is unstable precisely when the curvature zero set changes topology.
The same curvature criterion applies uniformly once the splitting condition on the reduced dynamics is verified.
Stability in the four-dimensional Lagrange configuration can be compared with its planar counterpart through the same framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to other Hamiltonian systems whose linearizations admit a similar symplectic splitting.
Numerical checks on specific four- and five-body relative equilibria could test whether the topology change remains a reliable marker beyond the three-body case.
A geometric reformulation of Moeckel's conjecture on central configurations might become feasible if the curvature zero set can be computed directly from the potential.
The ambient-dimension study suggests that higher-dimensional gravity models could exhibit qualitatively different stability patterns detectable by the same curvature diagnostic.

Load-bearing premise

The electromagnetic curvature model faithfully reproduces the stability properties of the original Newtonian n-body dynamics without adding or removing stability through the analogy itself.

What would settle it

A concrete relative equilibrium in the planar n-body problem whose reduced dynamics splits on invariant symplectic planes, whose stability status is independently known, yet whose electromagnetic curvature zero set shows no topology change at the expected threshold.

Figures

Figures reproduced from arXiv: 2604.07975 by Giorgia Testolina, Luca Asselle.

**Figure 2.** Figure 2: If √ α + √ β < √ 2, the positive curvature region is delimited by a hyperbola [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: If √ α + √ β > √ 2, the positive curvature region is delimited by an ellipse. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

read the original abstract

In this paper we study the linear stability of relative equilibria in the Newtonian $n$-body problem from the viewpoint of electromagnetic systems. We first examine the effect of the ambient dimension on stability, starting from the Lagrange equilateral triangle solutions of the three-body problem in $\mathbb R^4$. We then initiate a new approach to stability based on electromagnetic curvature. In a two-dimensional model, we relate linear stability to both the Ma\~n\'e critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold. This criterion is then applied to the planar $n$-body problem: in the three-body case, we recover Routh's classical criterion, and, more generally, we obtain an instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes. These results suggest a new geometric perspective on linear stability and on questions related to Moeckel's conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The electromagnetic curvature approach recovers Routh's criterion for three bodies but its reliability for general n-body stability hinges on unverified fidelity of the analogy.

read the letter

Here's the core of it: this paper uses an electromagnetic systems viewpoint to study linear stability of relative equilibria in the Newtonian n-body problem. They start by checking how the ambient space dimension influences stability, with examples like the Lagrange solutions in four dimensions. Then they develop a criterion based on electromagnetic curvature in a two-dimensional model, linking stability to the Mañé critical value and changes in the topology of the curvature's zero set. Applying this to the planar n-body problem, they get back Routh's classical criterion for three bodies and propose an instability criterion for cases where the reduced dynamics splits along invariant symplectic planes. The new element is this curvature topology idea as a way to detect stability thresholds. It performs well in reproducing the known three-body result, which lends some credibility to the framework. The geometric perspective could be useful for thinking about broader questions like Moeckel's conjecture on stability. On the downside, the key assumption is that this electromagnetic model mirrors the Newtonian linear stability exactly, without introducing its own effects from the potential or curvature definition. The concern about possible artifacts in the zero-set topology is worth taking seriously, because if the change doesn't precisely match the actual stability boundary, then the general instability test for split symplectic planes might not hold up as claimed. It would help to see more explicit verification that the symplectic structure and eigenvalues are preserved through the construction. This kind of work appeals to people in dynamical systems who are comfortable with symplectic geometry and geometric mechanics. It has enough substance and a clear new angle to warrant sending it out for peer review, though the authors should expect questions on the analogy's accuracy and perhaps some expansion on the general case.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a geometric approach to linear stability of relative equilibria in the Newtonian n-body problem by analogy with electromagnetic systems. It first studies ambient-dimension effects on stability for the Lagrange equilateral solutions in R^4, then introduces electromagnetic curvature in a 2D model that relates stability to the Mañé critical value and to topological changes in the zero set of the curvature. This criterion recovers Routh's classical result for the planar three-body problem and yields a general instability test when the reduced linearized dynamics splits along invariant symplectic planes, offering a perspective on Moeckel's conjecture.

Significance. If the electromagnetic-curvature construction faithfully reproduces Newtonian stability thresholds without model artifacts, the work supplies a new geometric tool for celestial mechanics. Recovery of Routh's criterion provides concrete evidence that the method can reproduce known results, and the splitting-based instability criterion could help address open questions such as Moeckel's conjecture on the stability of relative equilibria.

major comments (2)

[2D electromagnetic model section] The central claim that a topology change in the electromagnetic-curvature zero set marks the stability threshold rests on the unverified assumption that the 2D electromagnetic model introduces no extraneous artifacts relative to the Newtonian symplectic structure. Explicit verification is required that the zero-set transition coincides exactly with the known stability boundary for the Lagrange solutions (rather than arising from the specific choice of electromagnetic potential or curvature definition).
[planar n-body application] The general instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes requires a detailed check that the splitting preserves the original eigenvalues of the Newtonian linearization. Without this, the recovery of Routh's criterion could be coincidental rather than confirmatory of the method.

minor comments (2)

[Introduction] The definition of electromagnetic curvature and its relation to the Mañé critical value should be stated explicitly in the introduction before the 2D model is applied.
[Throughout] Notation for the reduced symplectic planes and the curvature zero set should be made consistent between the abstract, the 2D model, and the n-body application.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We respond point by point to the major comments, clarifying the verifications already present and indicating revisions that will be incorporated.

read point-by-point responses

Referee: [2D electromagnetic model section] The central claim that a topology change in the electromagnetic-curvature zero set marks the stability threshold rests on the unverified assumption that the 2D electromagnetic model introduces no extraneous artifacts relative to the Newtonian symplectic structure. Explicit verification is required that the zero-set transition coincides exactly with the known stability boundary for the Lagrange solutions (rather than arising from the specific choice of electromagnetic potential or curvature definition).

Authors: In the manuscript we carry out an explicit verification for the Lagrange equilateral solutions. We compute the electromagnetic curvature function directly in the 2D model as a function of the mass parameter and show that its zero set changes topology precisely when the mass ratio reaches the value at which the Newtonian linearization has a zero eigenvalue (Routh's critical ratio). This is obtained by solving the curvature expression and tracking the connected components of the zero set, which transition at the same parameter value as the classical stability boundary. The agreement indicates that the chosen electromagnetic potential reproduces the Newtonian threshold without introducing extraneous artifacts. We will revise the section to include the full algebraic steps of this computation together with a side-by-side comparison of the critical values. revision: yes
Referee: [planar n-body application] The general instability criterion for relative equilibria whose reduced linearized dynamics splits along invariant symplectic planes requires a detailed check that the splitting preserves the original eigenvalues of the Newtonian linearization. Without this, the recovery of Routh's criterion could be coincidental rather than confirmatory of the method.

Authors: The splitting arises from the invariant symplectic subspaces of the linearized Hamiltonian vector field at the relative equilibrium. Because each subspace is invariant under the linear flow and the restricted symplectic form remains nondegenerate, the characteristic polynomial of the full linearization is the product of the characteristic polynomials on the individual planes. Consequently the spectrum of the complete system is exactly the union of the spectra on the planes, with no eigenvalues added or removed. Instability detected by the curvature criterion on any single plane therefore implies instability of the full system. In the three-body case the reduced dynamics occupy a single effective plane, so the criterion recovers Routh's result directly rather than coincidentally. We will add a short lemma in the revised version that states this eigenvalue-preservation property explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops a geometric stability criterion for relative equilibria by relating linear stability in the Newtonian n-body problem to the Mañé critical value and topology of the electromagnetic curvature zero set in a 2D model. It recovers Routh's classical criterion for the planar 3-body problem as an application and derives an instability criterion for cases where reduced linearized dynamics split along invariant symplectic planes. These steps are presented as consequences of the electromagnetic analogy and curvature analysis rather than reductions to fitted parameters, self-definitions, or load-bearing self-citations. The recovery of a known result serves as external verification, and the central claims introduce independent geometric content without evident tautological equivalence to inputs. No steps match the enumerated circularity patterns based on the provided abstract and description.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard Hamiltonian and symplectic geometry assumptions for the n-body problem plus the novel electromagnetic analogy; no free parameters are evident from the abstract, and the curvature is introduced as part of the modeling rather than an independently evidenced entity.

axioms (2)

domain assumption The Newtonian n-body problem admits a Hamiltonian formulation with relative equilibria arising from rotational symmetry.
Invoked implicitly when discussing relative equilibria and their linearization in the abstract.
ad hoc to paper Electromagnetic systems provide a valid analogy for analyzing curvature effects on stability in the n-body context.
Central to the new approach described in the abstract.

invented entities (1)

electromagnetic curvature no independent evidence
purpose: To serve as a geometric indicator whose zero-set topology signals the linear stability threshold.
Introduced as the core of the new stability criterion; no independent falsifiable evidence outside the model is mentioned.

pith-pipeline@v0.9.0 · 5452 in / 1400 out tokens · 27329 ms · 2026-05-10T17:59:38.927140+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

In a two-dimensional model, we relate linear stability to both the Mañé critical value and to the behavior of the zero set of the electromagnetic curvature, highlighting a change in its topology at the stability threshold.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the linear stability condition √α + √β < √(2k) is equivalent to Routh’s classical criterion

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Morse theory forS-balanced configurations in the Newtoniann-body problem.J

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work page arXiv 2025
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Vivina Barutello, Riccardo Jadanza, and Alessandro Portaluri. Linear instability of relative equilibria forn-body problems in the plane.J. Differ. Equations, 257(6):1773–1813, 2014

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Kenneth R. Meyer and Dieter S. Schmidt. Elliptic relative equilibria in theN-body problem. J. Differ. Equations, 214(2):256–298, 2005

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Lectures on central configurations

Richard Moeckel. Lectures on central configurations. Lecture notes, Centre de Re- cerca Matemàtica, January 2014. URL:http://www.math.umn.edu/~rmoeckel/notes/ CentralConfigurations.pdf

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Gareth E. Roberts. On kite central configurations.Nonlinearity, 38(7):075001, July 2025. 20

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On the role of bifurcation theory and curvature in the existence and stability of periodic orbits in electromagnetic systems and then-body problem.Dissertation, 2025

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[1] [1]

Lectures on the free period Lagrangian action functional.J

Alberto Abbondandolo. Lectures on the free period Lagrangian action functional.J. Fixed Point Theory Appl., 13(2):397–430, 2013.doi:10.1007/s11784-013-0128-1

work page doi:10.1007/s11784-013-0128-1 2013

[2] [2]

Le problème desncorps et les distances mutuelles

Alain Albouy and Alain Chenciner. Le problème desncorps et les distances mutuelles. (french) [Then-body problem and mutual distances].Inventiones Mathematicae, 131(1):151–184, 1998

work page 1998

[3] [3]

Alain Albouy and Holger R. Dullin. Relative equilibria of the 3-body problem inR4.J. Geom. Mech., 12(3), 2020.doi:10.3934/jgm.2020012

work page doi:10.3934/jgm.2020012 2020

[4] [4]

Alain Albouy and Holger R. Dullin. Bounded orbits for three bodies inR4.Geometric Me- chanics, 01(01), 2024.doi:10.1142/S2972458924500035

work page doi:10.1142/s2972458924500035 2024

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The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles.J

Luca Asselle and Gabriele Benedetti. The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted cotangent bundles.J. Topol. Anal., 8(3):545–570, 2016.doi: 10.1142/S1793525316500205

work page doi:10.1142/s1793525316500205 2016

[6] [7]

Morse theory forS-balanced configurations in the Newtoniann-body problem.J

Luca Asselle and Alessandro Portaluri. Morse theory forS-balanced configurations in the Newtoniann-body problem.J. Dyn. Differ. Equations, 35:907–946, 2021

work page 2021

[7] [8]

Spectral stability, spectral flow and circular relative equilibria for the Newtoniann-body problem.J

Luca Asselle, Alessandro Portaluri, and Li Wu. Spectral stability, spectral flow and circular relative equilibria for the Newtoniann-body problem.J. Differ. Equations, 337:323–362, 2022

work page 2022

[8] [9]

Magnetic curvature and existence of a closed magnetic geodesic on low energy levels.Int

Valerio Assenza. Magnetic curvature and existence of a closed magnetic geodesic on low energy levels.Int. Math. Res. Not., 2024(21):13586–13610, 2024.doi:10.1093/imrn/rnae209

work page doi:10.1093/imrn/rnae209 2024

[9] [10]

Electromagnetic curvature via Jacobi-Maupertuis and beyond.arXiv preprint, arXiv:2510.27514, 2025

Valerio Assenza and Giorgia Testolina. Electromagnetic curvature via Jacobi-Maupertuis and beyond.arXiv preprint, arXiv:2510.27514, 2025

work page arXiv 2025

[10] [11]

Linear instability of relative equilibria forn-body problems in the plane.J

Vivina Barutello, Riccardo Jadanza, and Alessandro Portaluri. Linear instability of relative equilibria forn-body problems in the plane.J. Differ. Equations, 257(6):1773–1813, 2014

work page 2014

[11] [12]

Stability of relative equilibria and morse index of central configurations.C

Xijun Hu and Shanzhong Sun. Stability of relative equilibria and morse index of central configurations.C. R. Math. Acad. Sci. Paris, 347(2):1309–1312, 2009.doi:https://doi. org/10.1093/imrn/rnz246

work page doi:10.1093/imrn/rnz246 2009

[12] [13]

Meyer and Dieter S

Kenneth R. Meyer and Dieter S. Schmidt. Elliptic relative equilibria in theN-body problem. J. Differ. Equations, 214(2):256–298, 2005

work page 2005

[13] [14]

Linear stability of relative equilibria with a dominant mass.J

Richard Moeckel. Linear stability of relative equilibria with a dominant mass.J. Dyn. Differ. Equations, 6:37–51, 1994

work page 1994

[14] [15]

Lectures on central configurations

Richard Moeckel. Lectures on central configurations. Lecture notes, Centre de Re- cerca Matemàtica, January 2014. URL:http://www.math.umn.edu/~rmoeckel/notes/ CentralConfigurations.pdf

work page 2014

[15] [16]

Gareth E. Roberts. On kite central configurations.Nonlinearity, 38(7):075001, July 2025. 20

work page 2025

[16] [17]

On the role of bifurcation theory and curvature in the existence and stability of periodic orbits in electromagnetic systems and then-body problem.Dissertation, 2025

Giorgia Testolina. On the role of bifurcation theory and curvature in the existence and stability of periodic orbits in electromagnetic systems and then-body problem.Dissertation, 2025. doi:https://doi.org/10.13154/294-14236. 21

work page doi:10.13154/294-14236 2025