Lagrange points of the restricted three-body problem in spaces of constant curvature

Carlos Barrera-Anzaldo; Luis C. Garc\'ia-Naranjo; Miguel Ayala

Lagrange points of the restricted three-body problem in spaces of constant curvature

T0 review · 7 major / 0 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Positive curvature forces new Lagrange-type equilibria in the curved three-body problem and stabilizes the classical triangular points.

desk verdict Solid rigorous progress on the curved restricted three-body problem, but the introduction overclaims what Section 8 actually proves about stabilization of the triangular equilibria. read the letter →

arxiv 2607.19148 v1 pith:3BEVTGTF submitted 2026-07-21 math-ph math.MP

Miguel Ayala , Carlos Barrera-Anzaldo , Luis C. García-Naranjo This is my paper · ORCID

classification math-phmath.MP MSC 70F0737N0537J2037J2565G30

keywords N-bodyproblemincurvedspacesrestrictedthree-bodyLagrangepointsrelativeequilibriabifurcationstabilitycomputer-assistedproofsconstantcurvature

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the circular restricted three-body problem on surfaces of constant curvature, the curved analogue of the classical planar problem with five Lagrange points. Its central claim is that on a positively curved sphere the surface's topology imposes a rigid balance among equilibrium types: however the amended potential's critical points are counted, #maxima + #minima − #saddles must equal −2. Since the five planar Lagrange points alone do not satisfy this balance, extra relative equilibria must appear for small positive curvature; the paper identifies them as three collinear points, E2, E3, and A1, and rigorously proves their existence and stability for small mass ratios. It also proves that the triangular points L4 and L5 are linearly stable for a wider range of mass ratios when curvature is positive and unstable when it is negative. These results convert earlier numerical observations into rigorous analysis, with computer-assisted proofs covering a substantial region of the parameter plane.

What carries the argument

The central object is the amended potential V_{κ,μ} on the unit sphere (or pseudosphere), obtained by rescaling the curved problem; its critical points are in one-to-one correspondence with relative equilibria. The load-bearing identity is the index balance #maxima + #minima − #saddles = −2, derived from the classical index theorem for vector fields on a sphere after extending the gradient field over the four singular points (the two primaries and their antipodes). A second piece of machinery is the 'triangular-balanced configuration' condition sin(d₁) = Λ sin(d₂) (with a hyperbolic analogue for negative curvature), expressed in distance coordinates, which reduces the search for triangular e

What would settle it

A direct numerical count of all nondegenerate critical points of V_{κ,μ} on the sphere for a fixed μ ∈ (0,1) and κ ∈ (0,π²/4) that does not yield exactly #maxima + #minima − #saddles = −2 would refute the topological theorem; equivalently, a numerical continuation showing that E2, E3, or A1 persists (rather than unbinding to infinite distance) as κ → 0+ would falsify the corollary that new equilibria must appear and then disappear when the space flattens.

Watch

Extended reading notes

Core claim

On its own terms, the discovery is that the relative equilibria (the curved analogues of the Lagrange points) are governed by a topological balance condition: for fixed mass ratio μ ∈ (0,1) and curvature 0 < κ < π²/4, every nondegenerate critical point of the amended potential V_{κ,μ} obeys #maxima + #minima − #saddles = −2. The balance follows from the index theorem for vector fields on the sphere, with the two primaries and their antipodal points acting as four additional sources and sinks after the gradient flow is extended. The immediate corollary, that new relative equilibria must exist for small positive curvature, is made constructive: for small μ these are shown to be the collinear p

Load-bearing premise

The primaries are assumed to move on the acute circular relative-equilibrium branch of the two-body problem that connects smoothly to the planar circular solution, with their mutual distance fixed to 1; the other (obtuse) branch on the sphere is excluded, so the classification and the count apply only to that branch.

Editorial extensions

If this is right

For small positive curvature and small mass ratio, the curved restricted three-body problem has exactly six collinear relative equilibria: the continuations of L1, L2, L3 plus the new points E2, E3, A1; as κ → 0+ the new ones escape to infinite distance from the rotation center.
The new collinear point E2 is a local minimum of the potential and hence Lyapunov stable, while L1, L2, L3, E3, and A1 are saddle-type (center–saddles) in the proven parameter ranges.
For sufficiently small positive curvature, the only triangular relative equilibria are the continuations of L4 and L5, for every mass ratio μ ∈ (0,1).
Positive curvature stabilizes L4 and L5: the interval of mass ratios for which they are linearly (gyroscopically) stable is strictly larger than in the planar problem; negative curvature destabilizes them.
Asymptotic expansions as κ → 0 and μ → 0 determine the locations of all collinear equilibria, showing which ones converge to the planar Lagrange points and which ones diverge.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An immediate extension the authors leave implicit: the same index-balance argument should apply to the obtuse branch of the two-body problem on the sphere, yielding a separate count; completing that analysis would give a truly complete classification of relative equilibria on the sphere rather than only on the acute branch.
The index-balance identity is independent of the specific gravitational potential as long as the amended potential has the same attracting/repelling behavior at the four singular points; any interaction law with the same singularity structure on the sphere should obey the same −2 constraint, suggesting the count is geometric, not dynamical.
The stabilizing effect of positive curvature on L4 and L5 suggests a testable physical analogue: in a slightly curved model of a Sun–planet–massless-satellite system, the stability window for the triangular points should broaden relative to the Euclidean estimate; numerical integration of the full curved equations would settle whether the gyroscopic stabilization persists nonlinearly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real contribution, not just a numerical study dressed up. The analytic theorems are the meat: Theorem 4.1 gives a Poincaré–Hopf balance that forces new relative equilibria for κ>0, Theorem 5.2 classifies collinear RE for small κ with explicit parameter ranges, Theorem 6.2 proves uniqueness of triangular RE for small κ, and Theorem 7.2 establishes stability of the collinear RE. The CAPs extend these results to larger regions and are backed by cited Julia code using interval arithmetic. The asymptotic expansions in Section 9 cleanly connect the new equilibria to the planar limit. The topological explanation—compactness of the sphere forces extra equilibria—is genuinely novel and a nice addition.

Credit where it's due: the problem is formulated carefully, with the distance normalized to 1 so that κ and μ are the only parameters, and the stability criteria are derived rather than fitted. The paper is honest in the body: Conjectures 5.6, 6.4, 7.4, and 8.1 are explicitly labeled as conjectures, and Remark 2.3 clearly states the obtuse-branch exclusion.

The soft spot is the advertised 'positive curvature stabilizes L4 and L5.' The introduction says Section 8 'prove[s]' this, but Section 8 contains no analytic stability theorem for triangular RE. It offers CAPs on a subregion of P_{μ_T}, which are inconclusive near κ=0 and near bifurcations, along with numerical evidence and Conjecture 8.1. The stress-test note is correct here. This is not a fatal flaw—the core existence/stability theorems are proven—but the intro should be toned down to match the actual content. Minor issues: Theorem 6.2 is existential and gives no explicit κ*; there is a small typo in Appendix B.1 about endpoint behavior of f4; and the CAP code is cited without a commit hash. These are easy fixes.

Overall, the paper is a solid contribution to celestial mechanics and Hamiltonian dynamics, and it deserves a serious referee. I would recommend sending it to peer review, with a request that the authors revise the introduction and clarify the distinction between proven results and numerical evidence.

Referee Report

7 major / 0 minor

Summary. The paper formulates the restricted three-body problem on surfaces of constant curvature as an autonomous two-degree-of-freedom Lagrangian system whose only parameters are the mass ratio μ and the curvature κ, with the Riemannian distance between the primaries normalized to 1. For κ>0, the authors prove a Poincaré–Hopf balance relation for the critical points of the amended potential V_{κ,μ} (Theorem 4.1), which forces the existence of new relative equilibria beyond the continuations of L1,...,L5. They then give analytic existence and stability results for these equilibria for small μ and small κ>0 (Theorems 5.2, 6.2, 7.2), computer-assisted proofs (CAPs) on the parameter region P_{μ_T}, asymptotic expansions as κ→0 (Theorems 9.1 and 9.2), and numerical bifurcation analyses. Several full classifications are stated as Conjectures 5.6, 6.4, 7.4, and 8.1.

Significance. If the main results hold, the paper makes a valuable contribution to the curved N-body problem: it gives a rigorous topological explanation for the appearance of new relative equilibria in positive curvature, proves analytic continuation and stability for several branches, and supplies reproducible, machine-checked CAPs with cited code. The stability criteria in Propositions 4.11 and 4.12 are derived, not fitted, and the asymptotic expansions sharpen previous numerical observations. The main advertised conclusion—that positive curvature stabilizes L4 and L5—is, however, only partially supported by the rigorous results: the analytic theorems do not cover the triangular equilibria, and the CAPs are restricted to a subregion and are inconclusive near κ=0 and near bifurcations. This gap must be addressed before the stabilization claim can be accepted as proven.

major comments (7)

[§6, Theorem 6.2] The Introduction states that Section 8 'prove[s] that positive curvature has a stabilizing effect for L4 and L5', and §1.1 repeats this claim. But Section 8 contains no analytic stability theorem for triangular RE. It offers CAPs on a subregion of P_{μ_T} (Fig. 15(a),(c)), explicitly inconclusive near κ=0 and near bifurcations, numerical panels, and Conjecture 8.1. There is no rigorous proof that the elliptic/gyroscopically-stabilized region extends to all κ>0 for any fixed μ, nor a quantitative statement as κ→0. The abstract's weaker language ('indicates') is appropriate; the Introduction's 'prove' is not. Please either strengthen the proof or revise the claim to 'provide CAP-verified and numerical evidence'.
[§1.2, §5.3–5.5, §6.2–6.3] Theorem 6.2 asserts existence of exactly two triangular RE for sufficiently small κ>0, but gives no explicit κ* and no dependence on μ. The proof in Appendix B.2 also does not produce a constructive bound. As a result, the theorem cannot be combined with the CAPs (which are validated only for κ≥5.075×10^{-5}) to prove that the 'two triangular RE' found by CAPs are continuations of L4 and L5; this identification is left as a conjecture (see the discussion after Fig. 10). Since this identification is used in the stability conclusions of Section 8, the missing quantitative threshold is load-bearing for the claimed stabilization story.
[Remark 2.3] The paper consistently distinguishes proved results from conjectures, which is commendable. However, the abstract and introduction describe the work as providing classification results for the curved R3BP, while the full classification in the main parameter region P_{μ_T} rests on Conjectures 5.6, 6.4, 7.4, and 8.1. Remark 5.7 explicitly admits that the existence proofs for parts of P_{μ_T} and the bifurcation curves are missing. This is not a flaw in the proven theorems, but the manuscript should state more prominently, in the abstract and introduction, that the complete classification in P_{μ_T} is partly conjectural, with rigorous results covering a substantial subregion.
[Appendix C.2] All positive-curvature results assume the primaries move on the acute circular relative-equilibrium branch, with the obtuse branch explicitly excluded. This is a legitimate scope choice, and it is clearly flagged in Remark 2.3. Nevertheless, because the abstract's 'spaces of constant curvature' could be read as covering the whole sphere, the limitation should be echoed in the abstract or the opening of the introduction. The current phrasing in §1.2 mentions the acute-angle restriction only in a parenthetical about circular motions.
[Footnotes 3 and 10] In the first paragraph of Appendix C.2, the text refers to 'Propositions 4.3.1 or 4.15'. Proposition 4.3.1 does not exist; the intended references are presumably Proposition 4.11 (collinear) and Proposition 4.12 or 4.15 (triangular). This typo should be corrected, along with the surrounding cross-reference style.
[§1.3, §2.3, §5.4] The abuse of notation V_{\kappa,\mu} to denote potentials on both M^\pm and S_\kappa is used heavily; in Eq. (3.9) the potential is defined on the rescaled space, but in later sections the same symbol is used for the physical potential. This is not a mathematical error but makes the text harder to follow; a short notational remark would help.
There are several typographical issues: 'whlie' in §1.3, 'attaraction' in §2.3, and 'F or' at the start of items in Conjectures 5.6, 6.4, 7.4, and 8.1. The reference [2] also has a doubled slash in the URL. These do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular step satisfies the quote-and-reduction test; the derivation is self-contained from the explicit Lagrangian and potential, with only upstream two-body self-citations used as modelling input.

full rationale

The central derivation chain is not circular. The amended potential V_{κ,μ} is obtained by explicit substitution from the Lagrangian (3.8)-(3.9), and RE are characterized as critical points (Prop. 3.1, Appendix A). The collinear and triangular existence conditions (4.8), (4.21) are computed derivatives; Theorems 5.2 and 6.2 are proved by root counting and sign/concavity estimates in Appendices B.1-B.2; and the stability statements follow from linearization and the Hessian/eigenvalue criteria of Propositions 4.11-4.12 and Appendix A, not from fitted values. Theorem 4.1's Poincaré-Hopf balance uses the stated limits (4.2) of the same explicit potential, and Corollary 4.2 derives the existence of additional RE from that index count rather than assuming them. The CAPs use interval arithmetic to enclose true values, are explicitly inconclusive near κ=0 and near bifurcations, and the paper openly labels unproved parts as Conjectures 5.6, 6.4, 7.4, 8.1 and Remark 5.7. Citations to the authors' own prior two-body work [8,17,18] supply the circular-orbit input for the primaries, but the paper reproduces the explicit formulas (2.9)-(2.12) and notes they can be verified by direct calculation; this is an upstream modelling premise, not a recycled conclusion of the restricted-problem results. The exclusion of the obtuse two-body branch is an explicit scope limitation (Remark 2.3), not a self-citation used to forbid alternatives. The Introduction's claim that Sec. 8 'prove[s]' positive-curvature stabilization is stronger than the CAP-verified subregion plus numerical evidence and Conjecture 8.1, but that is an overclaim/correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the curved-two-body primary motion, the cot_κ potential model, topological index theory, and the reliability of the interval-arithmetic CAPs. There are no empirical free parameters and no new physical entities; the newly found equilibria are derived solutions, not postulated objects.

assumptions (5)

domain assumption Primaries move on the acute circular RE branch of the curved two-body problem with distance normalized to 1; the obtuse branch is excluded.
Invoked in Section 2.3 and Remark 2.3; this is what fixes κ<π²/4 and the family of primary motions (2.9).
domain assumption Gravitational interaction is modeled by the generalized cot_κ potential (2.7) with singularities at collision and, for κ>0, antipodal points.
This is the standard curved-space generalization from the literature, not derived in the paper.
standard math Poincaré-Hopf index theorem applies to the extended gradient vector field X on S², with the four singularities contributing index +1 each.
Used in the proof of Theorem 4.1; the index sum is the Euler characteristic 2.
domain assumption All critical points of V_{κ,μ} are non-degenerate in the regions where the topological count is applied.
Explicit hypothesis of Theorem 4.1; the CAPs verify non-degeneracy in the regions they cover, but the theorem itself is conditional.
domain assumption Interval arithmetic with directed rounding gives guaranteed enclosures, and the CAP code faithfully implements the paper's formulas.
The CAP methodology in Appendix C depends on this tooling assumption; no formal proof certificate is supplied.

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Pith. "Pith review of Lagrange points of the restricted three-body problem in spaces of constant curvature." pith.science (2026). https://pith.science/paper/3BEVTGTF

@misc{pith2026260719148,
  author       = {Pith},
  title        = {Pith review of: Lagrange points of the restricted three-body problem in spaces of constant curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BEVTGTF}},
  note         = {Machine review of arXiv:2607.19148}
}

abstract

We continue the study initiated by Kilin ({\em Reg. Chaot. Dyn.} 4, (1999)) and by Mart\'inez and Sim\'o ({\em Celest. Mech. Dynam. Astronom.} 128, (2017)) on the classification and stability of the relative equilibria of the restricted three-body problem in two-dimensional spaces of constant curvature, which generalize the classical Lagrange points of the planar problem. After formulating the problem as an autonomous Lagrangian system with two degrees of freedom, whose only parameters are the curvature $\kappa$ and the mass ratio $\mu$ of the primaries, we establish several classification results for the case $\kappa>0$ by combining analytical methods with computer-assisted proofs. These results provide rigorous confirmation of phenomena for which previously only numerical evidence was available. We also provide topological explanations for the qualitative differences between the behavior of relative equilibria in positive curvature and that observed in the planar and negative-curvature cases. Our analysis focuses on the regime of small $\mu$ and indicates that positive curvature has a stabilizing effect on the triangular equilibria $\Ll_4$ and $\Ll_5$, whereas negative curvature appears to have the opposite effect.

Figures

Figures reproduced from arXiv: 2607.19148 by the authors.

**Figure 1.** Lagrange points L1, . . . , L5 for the R3BP. These points correspond to critical points of the augmented potential V0,µ given in Eq. (2.16), projected onto the plane whose origin coincides with the center of mass of the primaries µ1, µ2. Some level curves of V0,µ are plotted in gray. In Sec. 2.4 below, we will give explicitly the Lagrangian system associated with the R3BP on a useful Riemannian model of the plane R … view at source ↗

**Figure 2.** The space M± for positive and negative curvature. The geodesic G ± connecting the primaries p1 and p2 is represented as a dashed curve which contains the center of rotation C, and, in the case of positive curvature, the antipodal points aj := −pj , j = 1, 2. The curve G ± divides M± into two regions and for the analysis of triangular RE we restrict to M± >0 , which is shaded in a darker color and contains the satel… view at source ↗

**Figure 3.** (a) Bounded domain Wκ for the distance coordinates, and (b) Half-space (S 2 1)>0. The points pj , j = 1, 2 are the primaries and aj := −pj , j = 1, 2 are their antipodal points. Their avatars in the domain Wκ are indicated as p˜j and a˜j . γκ,µ and ˜γκ,µ are the curves of triangle-balanced configurations in the respective domain. Proposition 4.8. The point r ∈ (S 2 1 )>0 with distance coordinates (d1, d2) ∈ Wk is a … view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: (a) Unbounded domain Wκ for the distance coordinates, and (b) Halfspace (L 2 1)>0. The points pj , j = 1, 2 are the primaries. Their avatars in the domain Wκ are indicated as p˜j . The curve of triangle-balanced configurations is respectively indicated by γκ,µ and ˜γκ…

**Figure 5.** Figure 5: The geodesic G + containing the rotation center C, the primaries pi (red points), and their antipodal points ai (gray points), i = 1, 2. These latter four points partition G + into the four segments I1, I2, I3, I4. According to our conventions C ∈ I1 and corresponds to…

**Figure 6.** Figure 6: The collinear RE in Theorem 5.2. The proof of Theorem 5.2 consists of estimating the number of roots of the functions θ 7→ fj (θ; κ, µ), j = 1, . . . , 4, as functions of the parameters (κ, µ), using techniques from elementary calculus. The details are provided in Sec.…

**Figure 7.** Figure 7: CAPs for the existence of collinear RE in the parameter region [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

**Figure 8.** Figure 8: Left panel: Numerical results for the existence of collinear relative equilibria [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]

**Figure 9.** Figure 9: shows a bifurcation diagram, obtained numerically, of the collinear RE for a fixed value of µ ∈ (0, µT) as a function of κ. In the figure, the vertical axis represents the signed Riemannian distance D to the center of rotation C measured in trigonometric sense accordin…

**Figure 10.** Figure 10: Number of triangular RE as a function of the parameters ( [PITH_FULL_IMAGE:figures/full_fig_p033_10.png]

**Figure 11.** Figure 11: Triangular relative equilibria. The value [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]

**Figure 12.** Figure 12: CAPs of stability of collinear relative equilibria in terms of the parameters [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]

**Figure 13.** Figure 13: Numerical evidence for the stability and instability regions in parameter [PITH_FULL_IMAGE:figures/full_fig_p037_13.png]

**Figure 14.** Figure 14: Numerical investigation of stability of the triangular RE L [PITH_FULL_IMAGE:figures/full_fig_p039_14.png]

**Figure 15.** Figure 15: Stability CAPs (panels (a), (c)) and numerical investigations (panels (b), [PITH_FULL_IMAGE:figures/full_fig_p040_15.png]

**Figure 16.** Figure 16: Linear stability of L4 and L5 as a function of the parameters κ and µ, for positive and negative curvature. 9 Asymptotic behavior of relative equilibria for small κ and µ. We now provide asymptotic expansions that yield information about the locations of the RE of the…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Lagrange points of the restricted three-body problem in spaces with constant curvature

[1]Andrade, J., P ´erez-Chavela, E., and Vidal, C.The restricted three body problem on surfaces of constant curvature.J. Differential Equations 265, 9 (2018), 4486–4529. [2]Ayala, M., Barrera-Anzaldo, C., and Garc ´ıa-Naranjo, L. C.Support codes for “Lagrange points of the restricted three-body problem in spaces with constant curvature”.https://github.com...

2018

[2] [6]

V., Garc ´ıa-Naranjo, L

67 [8]Borisov, A. V., Garc ´ıa-Naranjo, L. C., Mamaev, I. S., and Montaldi, J.Reduction and relative equilibria for the two-body problem on spaces of constant curvature.Celestial Mech. Dynam. Astronom. 130, 6 (2018), Paper No. 43,

2018

[3] [11]

M.On the stability of Lagrange’s periodic solutions of the restricted three-body problem.Dokl

[26]Leontovi ˇc, A. M.On the stability of Lagrange’s periodic solutions of the restricted three-body problem.Dokl. Akad. Nauk SSSR 143, 3 (1962), 525–528. 68 [27]Maciejewski, A. J., and Przybylska, M.Non-integrability of restricted two body problems in constant curvature spaces.Regul. Chaotic Dyn. 8, 4 (2003), 413–430. [28]Markeev, A. P.On the stability o...

1962

[4] [20]

C., Marrero, J

[17]Garc ´ıa-Naranjo, L. C., Marrero, J. C., P ´erez-Chavela, E., and Rodr ´ıguez-Olmos, M. Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2.J. Differential Equations 260, 7 (2016), 6375–6404. [18]Garc ´ıa-Naranjo, L. C., and Montaldi, J.Attracting and Repelling 2-Body Problems on a Family...

2016

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