Relative Equilibria for Scaling Symmetries and Central Configurations

Giovanni Rastelli; Manuele Santoprete

arxiv: 2408.15191 · v2 · submitted 2024-08-27 · 🧮 math-ph · math.DG· math.MP· math.SG· physics.class-ph

Relative Equilibria for Scaling Symmetries and Central Configurations

Giovanni Rastelli , Manuele Santoprete This is my paper

Pith reviewed 2026-05-23 21:23 UTC · model grok-4.3

classification 🧮 math-ph math.DGmath.MPmath.SGphysics.class-ph

keywords scaling symmetriesconformal momentum maprelative equilibriacentral configurationsn-body problemsymplectic geometryaugmented Hamiltonianconformally symplectic actions

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The pith

Scaling symmetries on exact symplectic manifolds define relative equilibria via a conformal augmented Hamiltonian that recovers the classical central configurations.

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

The paper develops a framework for scaling symmetries by introducing conformally symplectic maps and group actions on exact symplectic manifolds equipped with a primitive one-form. This leads to a conformal momentum map that generalizes the standard one and supports a generalized Noether theorem. Relative equilibria are characterized as critical points of an augmented Hamiltonian built from this map. The same construction on cotangent bundles produces an augmented potential whose critical points define central configurations for scaling-symmetric mechanical systems. Specializing the entire setup to the Newtonian n-body problem reproduces the familiar algebraic conditions that locate the classical central configurations.

Core claim

Relative equilibria of scaling symmetries are solutions to equations involving the conformal augmented Hamiltonian H_ξ and the primitive one-form θ. On cotangent bundles these reduce to equations involving an augmented potential U_ξ and the Lagrangian one-form. When the general theory is applied to the Newtonian n-body problem the resulting equations coincide exactly with the classical central-configuration equations.

What carries the argument

The conformal momentum map, which assigns to each infinitesimal scaling symmetry a function on the manifold that is used to construct the augmented Hamiltonian whose critical points locate the relative equilibria.

If this is right

Relative equilibria for scaling symmetries obey equations that differ in form and properties from those arising in ordinary symplectic actions.
Explicit formulas for the conformal momentum map are obtained once the action is lifted to the cotangent bundle via the scaled cotangent lift.
Central configurations are defined for any Hamiltonian system on a cotangent bundle that admits scaling symmetries, not only the Newtonian case.
The augmented potential U_ξ supplies the explicit stationarity condition for relative equilibria of simple mechanical systems with scaling symmetry.

Where Pith is reading between the lines

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

The same construction could be used to locate relative equilibria in other Hamiltonian systems that possess a global scaling symmetry, such as certain molecular or plasma models.
Because the conformal momentum map produces a different conserved quantity, stability criteria derived from the augmented Hamiltonian may differ from those obtained with the ordinary momentum map.
The framework supplies a template for treating time-dependent or position-dependent scalings once the exactness assumption is relaxed in a controlled way.

Load-bearing premise

The symplectic manifold must be exact, with its two-form equal to the exterior derivative of a fixed primitive one-form.

What would settle it

An explicit calculation for the three-body problem in which the derived critical-point equations do not reduce to the known Euler or Lagrange central-configuration conditions would falsify the recovery claim.

read the original abstract

In this paper, we explore scaling symmetries within the framework of symplectic geometry. We focus on the action $\Phi$ of the multiplicative group $G = \mathbb{R}^+$ on exact symplectic manifolds $(M, \omega,\theta)$, with $\omega = -d\theta$, where $ \theta $ is a given primitive one-form. Extending established results in symplectic geometry and Hamiltonian dynamics, we introduce conformally symplectic maps, conformally Hamiltonian systems, conformally symplectic group actions, and the notion of conformal invariance. This framework allows us to generalize the momentum map to the conformal momentum map, which is crucial for understanding scaling symmetries. Additionally, we provide a generalized Hamiltonian Noether's theorem for these symmetries. We introduce the (conformal) augmented Hamiltonian $H_{\xi}$ and prove that the relative equilibria of scaling symmetries are solutions to equations involving $ H _{ \xi } $ and the primitive one-form $\theta$. We derive their main properties, emphasizing the differences from relative equilibria in traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive explicit formulas for the conformal momentum map. We also provide a general definition of central configurations for Hamiltonian systems on cotangent bundles that admit scaling symmetries. Applying these results to simple mechanical systems, we introduce the augmented potential $U_{\xi}$ and show that the relative equilibria of scaling symmetries are solutions to an equation involving $ U _{ \xi } $ and the Lagrangian one-form $\theta_L$. Finally, we apply our general theory to the Newtonian $n$-body problem, recovering the classical equations for central configurations.

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 paper builds a conformal extension of momentum maps and relative equilibria for scaling symmetries on exact symplectic manifolds, then recovers the standard central configuration equations for the n-body problem.

read the letter

The core contribution is the conformal momentum map and the associated notion of conformally Hamiltonian systems for actions of the multiplicative group on exact symplectic manifolds. They also give explicit formulas for the scaled cotangent lift and a generalized definition of central configurations that reduces to the usual one when applied to the Newtonian n-body problem on the cotangent bundle minus collisions. That recovery is the main check on the framework, and the abstract states it works. The setup is clean: they start from the exact symplectic structure, introduce the conformal augmented Hamiltonian, and derive the stationarity condition involving both that Hamiltonian and the primitive one-form. For simple mechanical systems they translate it into an equation with the augmented potential and the Lagrangian one-form. This is a direct, incremental extension of standard cotangent-bundle symplectic geometry rather than a reinvention. The main soft spot is the stress-test point on the primitive one-form: the relative-equilibrium condition keeps explicit dependence on θ, so the claim that it specializes exactly to the classical gradient condition on the Newtonian potential requires that all θ-dependent terms cancel after the scaled lift is substituted. The abstract asserts recovery but does not show the term-by-term cancellation, which would need to be verified in the full text. If that cancellation holds without extra assumptions, the work is internally consistent; if not, the generalization would need adjustment. The paper is aimed at researchers in celestial mechanics and symplectic reduction who already know the classical central-configuration equations and want a systematic way to include scaling symmetries. A reader in that subfield can extract the new definitions and the recovery result without much overhead. It is formally grounded enough and recovers enough known cases to merit sending to a referee rather than desk rejection, even if the proofs need tightening on the cancellation step.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a framework for scaling symmetries on exact symplectic manifolds (M, ω, θ) with ω = −dθ, introducing conformally symplectic actions, a conformal momentum map, and a conformal augmented Hamiltonian H_ξ. It proves properties of relative equilibria for these symmetries, specializes the setup to cotangent bundles via a scaled cotangent-lifted action, defines central configurations in this context, and applies the theory to the Newtonian n-body problem on T^*(ℝ^{3n} ∖ Δ), claiming to recover the classical central-configuration equations.

Significance. If the derivations hold without residual discrepancies, the work supplies a geometric unification of scaling symmetries and central configurations that extends standard symplectic reduction techniques. The explicit formulas for the conformal momentum map on cotangent bundles and the reduction to the augmented potential U_ξ constitute the main technical contribution.

major comments (1)

[Application to Newtonian n-body problem (final section)] The central claim in the final section—that the general relative-equilibrium condition involving H_ξ and θ recovers exactly the classical central-configuration equations for the Newtonian n-body problem—requires explicit term-by-term verification. The stationarity condition necessarily retains dependence on the primitive one-form θ (or θ_L); the manuscript must demonstrate that all such terms cancel identically upon substitution of the scaled cotangent lift and the Newtonian potential, or else identify any unmatched contributions.

minor comments (2)

[Section introducing conformal momentum map] Notation for the conformal momentum map and the augmented Hamiltonian H_ξ should be introduced with a clear comparison table to the standard momentum map to aid readability.
[Generalized Hamiltonian Noether's theorem] The statement of the generalized Noether theorem for conformal invariance would benefit from an explicit statement of the conserved quantity in terms of the conformal momentum map.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the final section. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Application to Newtonian n-body problem (final section)] The central claim in the final section—that the general relative-equilibrium condition involving H_ξ and θ recovers exactly the classical central-configuration equations for the Newtonian n-body problem—requires explicit term-by-term verification. The stationarity condition necessarily retains dependence on the primitive one-form θ (or θ_L); the manuscript must demonstrate that all such terms cancel identically upon substitution of the scaled cotangent lift and the Newtonian potential, or else identify any unmatched contributions.

Authors: We agree that the final section would benefit from an explicit term-by-term verification to make the cancellation of θ-dependent terms fully transparent. In the revised version we will insert a dedicated calculation (new subsection or appendix) that substitutes the scaled cotangent-lifted action and the Newtonian potential into the stationarity condition for H_ξ and shows, line by line, that every contribution involving θ (or θ_L) vanishes identically, thereby recovering the classical central-configuration equations without residual terms. revision: yes

Circularity Check

0 steps flagged

No circularity; general symplectic framework recovers classical central configurations without reduction to inputs

full rationale

The derivation introduces conformally symplectic structures and a conformal momentum map on exact symplectic manifolds (M, ω = −dθ), then defines relative equilibria via the augmented Hamiltonian H_ξ and applies the scaled cotangent lift to the Newtonian n-body problem on T^*(ℝ^{3n} ∖ Δ). The abstract states that this recovers the classical central-configuration equations. No quoted step equates a new prediction to a fitted parameter or prior self-citation by construction; the θ-dependent terms are part of the general setup and the recovery claim is presented as a verification of the framework rather than an identity forced by definition. The chain is self-contained against external benchmarks in symplectic geometry and classical mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard properties of exact symplectic manifolds and cotangent bundles; no free parameters or invented physical entities are introduced.

axioms (1)

domain assumption The symplectic form is exact: ω = −dθ for a given primitive one-form θ
Explicitly stated as the setup for the manifolds on which the scaling action Φ of R+ is defined.

pith-pipeline@v0.9.0 · 5833 in / 1070 out tokens · 23285 ms · 2026-05-23T21:23:10.066087+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

364, American Math- ematical Soc., 2008

Ralph Abraham and Jerrold E Marsden, Foundations of mechanics , no. 364, American Math- ematical Soc., 2008

work page 2008
[2]

16, 1966, pp

Vladimir Arnold, Sur la g´ eom´ etrie diﬀ´ erentielle des groupes de lie de dimension inﬁnie et ses applications ` a l’hydrodynamique des ﬂuides parfaits , Annales de l’institut Fourier, vol. 16, 1966, pp. 319–361

work page 1966
[3]

43, 435203

Alessandro Bravetti, Connor Jackman, and David Sloan, Scaling symmetries, contact reduc- tion and Poincar´ e’s dream, Journal of Physics A: Mathematical and Theoretical 56 (2023), no. 43, 435203

work page 2023
[4]

5, 978–1049

Renato C Calleja, Alessandra Celletti, and Rafael De la Llave, A KAM theory for conformally symplectic systems: eﬃcient algorithms and their validati on, Journal of Diﬀerential Equations 255 (2013), no. 5, 978–1049

work page 2013
[5]

Cari˜ nena, Fernando Falceto, and Manuel F

Jos´ e F. Cari˜ nena, Fernando Falceto, and Manuel F. Ra˜ nada, Canonoid transformations and master symmetries , Journal of Geometric Mechanics 5 (2013), no. 2, 151–166 (en)

work page 2013
[6]

3, 221–251

Robert L Devaney, Collision orbits in the anisotropic Kepler problem , Inventiones mathemat- icae 45 (1978), no. 3, 221–251

work page 1978
[7]

1, Springer Science & Business Media, 2012

Florin Diacu, Relative equilibria of the curved n-body problem , vol. 1, Springer Science & Business Media, 2012

work page 2012
[8]

228, American mathematical society, 2014

, Relative equilibria in the 3-dimensional curved n-body pro blem, vol. 228, American mathematical society, 2014

work page 2014
[9]

part i: Relative equilibria , Journal of nonlinear science 22 (2012), 247– 266

Florin Diacu, Ernesto P´ erez-Chavela, and Manuele Santoprete , The n-body problem in spaces of constant curvature. part i: Relative equilibria , Journal of nonlinear science 22 (2012), 247– 266

work page 2012
[10]

part ii: Singularities, Journal of nonlinear science 22 (2012), 267–275

, The n-body problem in spaces of constant curvature. part ii: Singularities, Journal of nonlinear science 22 (2012), 267–275

work page 2012
[11]

Florin Diacu, Cristina Stoica, and Shuqiang Zhu, Central conﬁgurations of the curved n-body problem, Journal of Nonlinear Science 28 (2018), 1999–2046

work page 2018
[12]

2, Springer, 1992

Manfredo Perdigao Do Carmo and J Flaherty Francis, Riemannian geometry, vol. 2, Springer, 1992

work page 1992
[13]

4, 420–427

Victor Dods and Corey Shanbrom, Numerical methods and closed orbits in the Kepler– Heisenberg problem, Experimental mathematics 28 (2019), no. 4, 420–427

work page 2019
[14]

, Self-similarity in the Kepler–Heisenberg problem , Journal of nonlinear science 31 (2021), 1–15

work page 2021
[15]

1, 139–152

Martin C Gutzwiller, The anisotropic Kepler problem in two dimensions , Journal of Mathe- matical Physics 14 (1973), no. 1, 139–152. 34

work page 1973
[16]

12, Oxford University Press, 2009

Darryl D Holm, Tanya Schmah, and Cristina Stoica, Geometric mechanics and symmetry: from ﬁnite to inﬁnite dimensions , vol. 12, Oxford University Press, 2009

work page 2009
[17]

35, Springer Science & Business Media, 2012

Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, vol. 35, Springer Science & Business Media, 2012

work page 2012
[18]

1, 121–130

Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry , Re- ports on mathematical physics 5 (1974), no. 1, 121–130

work page 1974
[19]

174, Cambridge University Press, 1992

Jerrold E Marsden, Lectures on mechanics , vol. 174, Cambridge University Press, 1992

work page 1992
[20]

17, Springer Science & Business Media, 2013

Jerrold E Marsden and Tudor S Ratiu, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems , vol. 17, Springer Science & Business Media, 2013

work page 2013
[21]

4, 276–300

Robert McLachlan and Matthew Perlmutter, Conformal Hamiltonian systems , Journal of Geometry and Physics 39 (2001), no. 4, 276–300

work page 2001
[22]

McLachlan and Christian Oﬀen, Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci , New Zealand Journal of Mathematics 48 (2018), 83–99

Robert I. McLachlan and Christian Oﬀen, Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci , New Zealand Journal of Mathematics 48 (2018), 83–99

work page 2018
[23]

Meyer and D.C

K.R. Meyer and D.C. Oﬃn, Introduction to Hamiltonian dynamical systems and the n-bo dy problem, Applied Mathematical Sciences, Springer International Publishing , 2017

work page 2017
[24]

James Montaldi, Relative equilibria and conserved quantities , Peyresq Lectures in Nonlinear Phenomena, World Scientiﬁc, 2000, p. 239

work page 2000
[25]

Richard Montgomery and Corey Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere , Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsd en (2015), 319–342

work page 2015
[26]

Manuele Santoprete, Gravitational and harmonic oscillator potentials on surfa ces of revolu- tion, Journal of mathematical physics 49 (2008), no. 4

work page 2008
[27]

, On the uniqueness of trapezoidal four-body central conﬁgur ations, Nonlinearity 34 (2021), no. 1, 424

work page 2021
[28]

David Sloan, Dynamical similarity , Phys. Rev. D 97 (2018), 123541

work page 2018
[29]

, Scale symmetry and friction , Symmetry 13 (2021), no. 9, 1639

work page 2021
[30]

Symmetry, Integrability and Geometry: Methods and Applications 17 (2021), 074

Tomasz Stachowiak, Andrzej J Maciejewski, et al., Non-integrability of the Kepler and the two-body problems on the Heisenberg group , SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 17 (2021), 074

work page 2021
[31]

I , Inventiones mathematicae 10 (1970), 305–331

Smale Stephen, Topology and mechanics. I , Inventiones mathematicae 10 (1970), 305–331

work page 1970
[32]

II: the planar n-body problem , Inventiones mathematicae 11 (1970), 45–64

, Topology and mechanics. II: the planar n-body problem , Inventiones mathematicae 11 (1970), 45–64

work page 1970
[33]

10, 6191–6225

Cristina Stoica, On the n-body problem on surfaces of revolution , Journal of Diﬀerential Equa- tions 264 (2018), no. 10, 6191–6225. 35

work page 2018
[34]

P-M Zhang, M Elbistan, PA Horvathy, and P Kosi´ nski, A generalized Noether theorem for scaling symmetry , The European Physical Journal Plus 135 (2020), 1–12. 36

work page 2020

[1] [1]

364, American Math- ematical Soc., 2008

Ralph Abraham and Jerrold E Marsden, Foundations of mechanics , no. 364, American Math- ematical Soc., 2008

work page 2008

[2] [2]

16, 1966, pp

Vladimir Arnold, Sur la g´ eom´ etrie diﬀ´ erentielle des groupes de lie de dimension inﬁnie et ses applications ` a l’hydrodynamique des ﬂuides parfaits , Annales de l’institut Fourier, vol. 16, 1966, pp. 319–361

work page 1966

[3] [3]

43, 435203

Alessandro Bravetti, Connor Jackman, and David Sloan, Scaling symmetries, contact reduc- tion and Poincar´ e’s dream, Journal of Physics A: Mathematical and Theoretical 56 (2023), no. 43, 435203

work page 2023

[4] [4]

5, 978–1049

Renato C Calleja, Alessandra Celletti, and Rafael De la Llave, A KAM theory for conformally symplectic systems: eﬃcient algorithms and their validati on, Journal of Diﬀerential Equations 255 (2013), no. 5, 978–1049

work page 2013

[5] [5]

Cari˜ nena, Fernando Falceto, and Manuel F

Jos´ e F. Cari˜ nena, Fernando Falceto, and Manuel F. Ra˜ nada, Canonoid transformations and master symmetries , Journal of Geometric Mechanics 5 (2013), no. 2, 151–166 (en)

work page 2013

[6] [6]

3, 221–251

Robert L Devaney, Collision orbits in the anisotropic Kepler problem , Inventiones mathemat- icae 45 (1978), no. 3, 221–251

work page 1978

[7] [7]

1, Springer Science & Business Media, 2012

Florin Diacu, Relative equilibria of the curved n-body problem , vol. 1, Springer Science & Business Media, 2012

work page 2012

[8] [8]

228, American mathematical society, 2014

, Relative equilibria in the 3-dimensional curved n-body pro blem, vol. 228, American mathematical society, 2014

work page 2014

[9] [9]

part i: Relative equilibria , Journal of nonlinear science 22 (2012), 247– 266

Florin Diacu, Ernesto P´ erez-Chavela, and Manuele Santoprete , The n-body problem in spaces of constant curvature. part i: Relative equilibria , Journal of nonlinear science 22 (2012), 247– 266

work page 2012

[10] [10]

part ii: Singularities, Journal of nonlinear science 22 (2012), 267–275

, The n-body problem in spaces of constant curvature. part ii: Singularities, Journal of nonlinear science 22 (2012), 267–275

work page 2012

[11] [11]

Florin Diacu, Cristina Stoica, and Shuqiang Zhu, Central conﬁgurations of the curved n-body problem, Journal of Nonlinear Science 28 (2018), 1999–2046

work page 2018

[12] [12]

2, Springer, 1992

Manfredo Perdigao Do Carmo and J Flaherty Francis, Riemannian geometry, vol. 2, Springer, 1992

work page 1992

[13] [13]

4, 420–427

Victor Dods and Corey Shanbrom, Numerical methods and closed orbits in the Kepler– Heisenberg problem, Experimental mathematics 28 (2019), no. 4, 420–427

work page 2019

[14] [14]

, Self-similarity in the Kepler–Heisenberg problem , Journal of nonlinear science 31 (2021), 1–15

work page 2021

[15] [15]

1, 139–152

Martin C Gutzwiller, The anisotropic Kepler problem in two dimensions , Journal of Mathe- matical Physics 14 (1973), no. 1, 139–152. 34

work page 1973

[16] [16]

12, Oxford University Press, 2009

Darryl D Holm, Tanya Schmah, and Cristina Stoica, Geometric mechanics and symmetry: from ﬁnite to inﬁnite dimensions , vol. 12, Oxford University Press, 2009

work page 2009

[17] [17]

35, Springer Science & Business Media, 2012

Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, vol. 35, Springer Science & Business Media, 2012

work page 2012

[18] [18]

1, 121–130

Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry , Re- ports on mathematical physics 5 (1974), no. 1, 121–130

work page 1974

[19] [19]

174, Cambridge University Press, 1992

Jerrold E Marsden, Lectures on mechanics , vol. 174, Cambridge University Press, 1992

work page 1992

[20] [20]

17, Springer Science & Business Media, 2013

Jerrold E Marsden and Tudor S Ratiu, Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems , vol. 17, Springer Science & Business Media, 2013

work page 2013

[21] [21]

4, 276–300

Robert McLachlan and Matthew Perlmutter, Conformal Hamiltonian systems , Journal of Geometry and Physics 39 (2001), no. 4, 276–300

work page 2001

[22] [22]

McLachlan and Christian Oﬀen, Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci , New Zealand Journal of Mathematics 48 (2018), 83–99

Robert I. McLachlan and Christian Oﬀen, Hamiltonian boundary value problems, conformal symplectic symmetries, and conjugate loci , New Zealand Journal of Mathematics 48 (2018), 83–99

work page 2018

[23] [23]

Meyer and D.C

K.R. Meyer and D.C. Oﬃn, Introduction to Hamiltonian dynamical systems and the n-bo dy problem, Applied Mathematical Sciences, Springer International Publishing , 2017

work page 2017

[24] [24]

James Montaldi, Relative equilibria and conserved quantities , Peyresq Lectures in Nonlinear Phenomena, World Scientiﬁc, 2000, p. 239

work page 2000

[25] [25]

Richard Montgomery and Corey Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere , Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsd en (2015), 319–342

work page 2015

[26] [26]

Manuele Santoprete, Gravitational and harmonic oscillator potentials on surfa ces of revolu- tion, Journal of mathematical physics 49 (2008), no. 4

work page 2008

[27] [27]

, On the uniqueness of trapezoidal four-body central conﬁgur ations, Nonlinearity 34 (2021), no. 1, 424

work page 2021

[28] [28]

David Sloan, Dynamical similarity , Phys. Rev. D 97 (2018), 123541

work page 2018

[29] [29]

, Scale symmetry and friction , Symmetry 13 (2021), no. 9, 1639

work page 2021

[30] [30]

Symmetry, Integrability and Geometry: Methods and Applications 17 (2021), 074

Tomasz Stachowiak, Andrzej J Maciejewski, et al., Non-integrability of the Kepler and the two-body problems on the Heisenberg group , SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 17 (2021), 074

work page 2021

[31] [31]

I , Inventiones mathematicae 10 (1970), 305–331

Smale Stephen, Topology and mechanics. I , Inventiones mathematicae 10 (1970), 305–331

work page 1970

[32] [32]

II: the planar n-body problem , Inventiones mathematicae 11 (1970), 45–64

, Topology and mechanics. II: the planar n-body problem , Inventiones mathematicae 11 (1970), 45–64

work page 1970

[33] [33]

10, 6191–6225

Cristina Stoica, On the n-body problem on surfaces of revolution , Journal of Diﬀerential Equa- tions 264 (2018), no. 10, 6191–6225. 35

work page 2018

[34] [34]

P-M Zhang, M Elbistan, PA Horvathy, and P Kosi´ nski, A generalized Noether theorem for scaling symmetry , The European Physical Journal Plus 135 (2020), 1–12. 36

work page 2020