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REVIEW 3 major objections 3 minor 11 references

Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere $\mathbb{M}^2_R$ for equal masses

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Pith's one-line read This paper claims that on a sphere, two equal masses at antipodal points can move on the geodesic circle as a limiting relative equilibrium, provided the equations are regularized at antipodal singularities.

desk verdict The new cot(θ/2) potential is a useful tweak, but the 'conjugated equilibrium solution' is built on a degenerate regularization and the main claim is vacuous. read the letter →

arxiv 1908.06011 v1 pith:67SBD6QP submitted 2019-08-14 physics.class-ph math-phmath.MP

Pedro P. Ortega Palencia , Guadalupe Reyes Victoria This is my paper · ORCID
classification physics.class-phmath-phmath.MP MSC 70F1553Z05
keywords two-dimensionalconformalspheretwo-bodyproblemrelativeequilibriaconjugatedpointsantipodalregularizedequationsisoscelesstereographicprojection
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 equal-mass two-body problem on a two-dimensional sphere and claims that antipodal (conjugated) points, traditionally singular for the standard cotangent potential, become accessible as limits of isosceles relative equilibria. The central move is to replace the usual potential, which blows up at antipodal points, with a modified potential that vanishes there and only keeps binary collisions as singularities. Starting from the isosceles equilibria $w_1(t)=\alpha e^{2it}$, $w_2(t)=-\alpha e^{2it}$, the paper lets $\alpha$ approach the sphere radius $R$. The limit is a pair of antipodal particles moving on the geodesic circle $|w|=R$ with velocity $-2iw$, which the paper names a conjugated equilibrium solution and shows satisfies a regularized version of the equations of motion, though not the original system. If the regularization is accepted, the result extends the known relative-equilibrium classification to the antipodal boundary while still recovering the Newtonian potential in the flat limit $R\to\infty$.

What carries the argument

The load-bearing object is the modified potential $U_R^{kj}=\frac{m_jm_k|R^2+w_k\bar w_j|}{R^2|w_k-w_j|}$, which uses $\cot(\theta/2)$ rather than $\cot\theta$: at antipodal points the factor $R^2+\bar w_j w_k$ vanishes instead of producing a singularity, so only binary collisions remain singular. The second mechanism is the regularized system (27), which clears the antipodal singularity by multiplying the equations of motion (9) by the vanishing factor $|R^2+\bar w_j w_k|$; it is in this regularized system that the limiting antipodal orbit is a solution. These two choices, the potential and the regularization, carry the entire argument, together with the isosceles family of relative equilibria that provides the limit.

What would settle it

Substitute $w_1(t)=\alpha e^{2it}$, $w_2(t)=-\alpha e^{2it}$ into the original system (9) and let $\alpha\to R$: the left-hand side of each equation tends to $0$, while the right-hand side tends to a nonzero vector of magnitude $m/(2R^2)$, so the limiting orbit does not satisfy (9). To test the regularized claim, note that at any antipodal pair $R^2+\bar w_2 w_1=0$, making both sides of (27) vanish identically; hence (27) is satisfied by every differentiable antipodal trajectory, and the specific circular motion is selected by the Killing-invariance and geodesic requirements, not by the equations themselves.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 3: for equal masses moving under the modified potential $U_R^{kj}=\frac{m_jm_k|R^2+w_k\bar w_j|}{R^2|w_k-w_j|}$, the isosceles relative equilibria $w(t)=(\alpha e^{2it}, -\alpha e^{2it})$ converge as $\alpha\to R$ to $z(t)=(R e^{2it}, -R e^{2it})$. This limiting configuration consists of two antipodal particles on the equator, both moving with velocity $-2iz_k$; the potential vanishes identically along it, the curve is a geodesic of $\mathbb{M}^2_R$, and the motion is invariant under the Killing vector field $\dot z=2iz$. The configuration is a solution of the regularized system (27), obtained by multiplying the original equations (9) by the vanishing factor $|R^2+\bar w_j w_k|$, and it is not a solution of the original system (9). The paper defines this limiting object as a conjugated equilibrium solution.

Load-bearing premise

The argument rests on treating the regularized system (27), obtained by multiplying the original equations by the vanishing factor $|R^2+\bar w_j w_k|$, as the correct continuation of the dynamics through antipodal points; without that choice the limiting orbit is not a solution of the original system, and the paper supplies no independent justification for preferring this particular regularization.

Editorial extensions

If this is right

  • With the modified potential, antipodal points are no longer singularities of the potential, so two equal masses can be placed exactly opposite each other and the equations acquire a well-defined limiting form there.
  • The isosceles relative equilibria have an explicit boundary case: as $\alpha\to R$, the motion becomes a pair of antipodal particles on the geodesic circle $|w|=R$, invariant under the Killing field $\dot z=2iz$.
  • The right-angled relative equilibria have a separate limit in which one particle sits at the origin and the other moves on the equator with velocity $-2iw$, as stated in Corollary 3.
  • Both the original and the modified potentials reduce to the Newtonian potential as $R\to\infty$, so the flat two-body problem is recovered from the sphere in the large-radius limit.
  • The conjugated equilibrium is a solution only of the regularized system (27), so the notion of relative equilibrium is extended to antipodal configurations in a regularized sense.

Reading between the lines

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

  • The paper does not point out that, because both sides of (27) vanish identically on any antipodal pair, the regularized equations alone do not determine the orbital motion; the 'equilibrium' status of the circular solution is really carried by the Killing invariance and the geodesic condition, not by a force balance.
  • The same regularization trick could be applied to other curved $n$-body potentials, so 'conjugated equilibrium' is tied to the choice of the vanishing factor, not to an invariant feature of the original dynamics.
  • A testable extension would be to check whether the limit survives for unequal masses or for $n>2$ equal masses: the algebraic system (26) would need new solutions, and any such antipodal configuration would again need to be tested against the original, unregularized equations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. This manuscript studies the two-body problem on the conformal sphere M_R^2 with a cotangent potential U_R = m_j m_k |R^2 + w_j \bar w_k|/(R^2 |w_j - w_k|). The authors derive equations of motion (9), characterize elliptic relative equilibria for the two-body equal-mass case, and then take the limit alpha to R of the isosceles family w(t) = (alpha e^{2it}, -alpha e^{2it}). The limit z(t) = (R e^{2it}, -R e^{2it}) places the two particles at antipodal points on the geodesic circle |w| = R. The paper defines this as a "conjugated equilibrium solution" by verifying the regularized system (27), obtained by multiplying (9) by |R^2 + \bar w_2 w_1|.

Significance. The proposed potential is attractive in that it is finite at antipodal configurations and reduces to the Newtonian potential as R tends to infinity, and the explicit limit computations in equations (23)-(25) are clear. However, the central construction is not sound as presented: the regularized system (27) is identically satisfied by every antipodal curve, so the main object introduced in Definition 1 is not a meaningful dynamical solution. The paper also contains algebraic inconsistencies in the relative-equilibrium equations. If the concept were redefined with a genuine, nondegenerate regularization, the result might be interesting, but the manuscript in its current form does not establish the existence or meaning of conjugated equilibrium solutions.

major comments (3)
  1. [5, Theorem 3 and Eqs. (23)-(25)] The regularized system (27) is degenerate at antipodal configurations. Whenever R^2 + \bar w_2 w_1 = 0, the left-hand side of the first equation contains the factor |R^2 + \bar w_2 w_1| = 0 and the right-hand side contains the factor R^2 + \bar w_2 w_1 = 0, so the equation reduces to 0 = 0; the second equation behaves identically. Hence every differentiable curve w(t) with w_2(t) = -R^2 / \bar w_1(t) satisfies (27), independently of its velocities or accelerations. The substitution z(t) = (R e^{2it}, -R e^{2it}) in Theorem 3 is therefore only a check of the identity 0 = 0, not a verification of an equilibrium condition. Because Definition 1 defines the conjugated equilibrium solution as a solution of this degenerate system, the central object of the paper is vacuous; the paper itself states that z(t) is not a solution of the original system (9).
  2. [5, Theorem 3 and Eqs. (23)-(25)] The limit argument does not establish an equilibrium. The potential (23) tends to zero along the limiting curve, and equation (8) shows that the force contains the quotient (R^2 + \bar w_j w_k) / |R^2 + \bar w_j w_k|, which has no well-defined value at the antipodal configuration. Invariance under the Killing vector field \dot w = 2 i w and the geodesic character of |w| = R are not sufficient conditions for a relative equilibrium of the two-body problem; one still needs a force balance. Since z(t) is not a solution of (9), the claim that the limit "behaves as a relative equilibrium" is unsupported.
  3. [3-4, Eqs. (11)-(15)] The algebraic derivation of the relative-equilibrium equations contains inconsistencies. Substituting \dot w_k = 2 i w_k into (9) yields 4 w_k (|w_k|^2 - R^2) / (R^2 + |w_k|^2) on the left, which after rearrangement gives 16 R^6 (|w_k|^2 - R^2) w_k / (R^2 + |w_k|^2)^3 and a single factor (R^2 + |w_j|^2) on the right; equations (11) and (14) instead display (R^2 + |w_k|^2)^4 in the denominator and (|w_j|^2 + R^2)^2 in the numerator of the summand. Equation (13) also has the opposite sign (R^2 - |w_{k,0}|^2). The proof of Corollary 2 claims that (14) follows from (13) by multiplication by e^{2it}, but such a multiplication cannot alter powers of the moduli. These discrepancies make the derivation of Theorem 2 unreliable as written, even if the classification itself is quoted from the literature.
minor comments (3)
  1. [2.1 and Eq. (8)] The statement that potential (6) "avoids the antipodal points as singularities" is misleading in view of (8), where |R^2 + \bar w_j w_k| appears in the denominator; the quotient (R^2 + \bar w_j w_k) / |R^2 + \bar w_j w_k| is undefined when the particles are antipodal.
  2. [5, Theorem 3 and Corollary 3] The velocity sign is inconsistent with the parametrization. For z_k(t) = R e^{2it} or z_k(t) = -R e^{2it}, the derivative is 2 i z_k(t), not -2 i z_k(t) as stated in both results.
  3. [Throughout] There are numerous typographical errors (e.g., "W e", "substituing", "hypotesis", "a solution for that real equation") and inconsistent notation, including the missing exponent in (16) where "|β|2" appears instead of "|β|^2".

Circularity Check

1 steps flagged · score 7.0 of 10

The 'conjugated equilibrium solution' is defined as a solution of a regularized system that vanishes identically on every antipodal curve; the verification in Theorem 3 reduces to 0=0.

  1. self definitional [Section 5, Theorem 3, Eq. (27), and Definition 1]
    "It is easy to see that the function (24) is not a solution of (9) but an other simple substitution shows that it satisfy the regularized system obtained of (9) when we avoid singularities due also to conjugated antipodal points, (27) ... We call to the solution (24) of the regularized system (27), invariant under the Killing vector field ˙z = 2iz a conjugated equilibrium solution"

    For the candidate z(t)=(R e^{2it}, -R e^{2it}), one has R^2+\bar z_2 z_1 = 0. In Eq. (27), the left-hand side is multiplied by |R^2+\bar z_2 z_1|=0 and the right-hand side contains the factor (R^2+\bar z_2 z_1)=0. Hence both sides vanish identically for any differentiable antipodal trajectory, independently of velocities, accelerations, or whether the curve is geodesic. The 'simple substitution' in Theorem 3 is therefore a tautology: (27) imposes no equation of motion at antipodal configurations. Definition 1 then defines the 'conjugated equilibrium solution' as a solution of a system that is vacuous on the entire class of antipodal curves, so the central object is introduced by construction rather than derived from dynamics.

full rationale

The geometric limit itself is real: as alpha → R, the isosceles relative equilibria w(t)=(alpha e^{2it}, -alpha e^{2it}) tend to the antipodal geodesic curve z(t)=(R e^{2it}, -R e^{2it}), and the potential (23) tends to zero. The circularity lies in the classification of this limit as an equilibrium. The regularized system (27) is obtained by multiplying the original system (9) by |R^2+\bar w_2 w_1| (and the symmetric factor for the second particle), and at the antipodal configuration this factor and the corresponding numerator factor both vanish. Consequently (27) reduces to 0=0 for every differentiable antipodal curve, not only for the particular geodesic chosen. Thus the statement that z(t) satisfies the regularized system is true by construction and does not constitute a dynamical verification. The paper acknowledges that (24) is not a solution of the original system (9), but it then defines the new object 'conjugated equilibrium solution' as a solution of the degenerate system, making the main claim definitionally forced. This is not a case of harmless self-citation or an independent external result; it is a self-definitional construction in which the predicted object satisfies the defining equations identically. The degree of circularity is high because the central result, the existence and nature of conjugated equilibrium solutions, reduces to the chosen form of the regularized system.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the newly introduced potential and the ad hoc regularized system, both chosen specifically to make antipodal configurations tractable. There are no fitted parameters. The standard mathematical background is the stereographic projection and the Killing vector field formalism.

assumptions (4)
  • ad hoc to paper The cotangent potential U_R = m_j m_k |R^2 + w_j bar{w}_k| / (R^2 |w_j - w_k|) is the correct interaction on the conformal sphere M^2_R.
    Introduced in Section 2.1 as a 'subtle variant' of the standard cot(theta) potential; its justification is that it avoids antipodal singularities, not derived from first principles.
  • ad hoc to paper The regularized system (27), obtained by multiplying (9) by |R^2 + bar{w}_j w_k|, is a valid regularization of the dynamics through antipodal points.
    Stated in Section 5 without derivation; the multiplication makes the antipodal configuration a trivial solution, so the regularization is definitional.
  • standard math The Killing vector field dot{w} = 2iw generates the one-parameter subgroup of SU(2) used to define elliptic relative equilibria.
    Standard fact in the geometric Erlangen program; used in Theorem 1 and Section 4.
  • standard math The stereographic projection formulas and trigonometric identities used to relate cot(theta/2) to the complex coordinates.
    Standard Riemann surface computations; used in equations (4)-(6).
invented entities (1)
  • Conjugated equilibrium solution
    purpose: Label for the limit of isosceles relative equilibria when both particles are antipodal on the geodesic circle |w|=R; it satisfies the regularized system (27) but not the original system (9).
    Defined in Definition 1; it is a solution only of the ad hoc regularized system, so it has no falsifiable handle outside this paper.

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Pith. "Pith review of Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere $\mathbb{M}^2_R$ for equal masses." pith.science (2026). https://pith.science/paper/67SBD6QP

@misc{pith2026190806011,
  author       = {Pith},
  title        = {Pith review of: Conjugated equilibrium solutions for the $2$--body problem in the two dimensional sphere $\mathbbM^2_R$ for equal masses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67SBD6QP}},
  note         = {Machine review of arXiv:1908.06011}
}
abstract

We study here the behaviour of solutions for conjugated (antipodal) points in the $2$-body problem on the two-dimensional sphere $\mathbb{M}^2_R$. We use a slight modification of the classical potential used commonly in \cite{Borisov}, \cite{Diacu} and \cite{Perez}, which avoids the conjugated (antipodal) points as singularities and permit us obtain solutions through these points, as limit of relative equilibria. Such limit solutions behave as relative equilibria because are invariant under Killing vector fields in the Lie Algebra ${\rm su} (2)$ and are geodesic curves.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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