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Compactness and Index of Ordinary Central Configurations for the Curved \(N\)-Body Problem

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Abstract

For the curved \(n\)-body problem, we show that the set of ordinary central configurations is away from singular configurations in \(\mathbb{H}^{3}\) with positive momentum of inertia, and away from a subset of singular configurations in \(\mathbb{S}^{3}\). We also show that each of the \(n!/2\) geodesic ordinary central configurations for \(n\) masses has Morse index \(n-2\). Then we get a direct corollary that there are at least \(\frac{(3n-4)(n-1)!}{2}\) ordinary central configurations for given \(n\) masses if all ordinary central configurations of these masses are nondegenerate.

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References

  1. Albouy, A. and Kaloshin, V., Finiteness of Central Configurations of Five Bodies in the Plane, Ann. of Math. (2), 2012, vol. 176, no. 1, pp. 535–588.

    Article  MathSciNet  Google Scholar 

  2. Arnol’d, V. I., Kozlov, V. V., and Neuıshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics,3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.

    Book  Google Scholar 

  3. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., Two-Body Problem on a Sphere: Reduction, Stochasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265–279.

    Article  MathSciNet  Google Scholar 

  4. Borisov, A. V., García-Naranjo, L. C., Mamaev, I. S., and Montaldi, M., Reduction and Relative Equilibria for the Two-Body Problem on Spaces of Constant Curvature, Celestial Mech. Dynam. Astronom., 2018, vol. 130, no. 6, 43, 36 pp.

    Article  MathSciNet  Google Scholar 

  5. Chenciner, A., The Angular Momentum of a Relative Equilibrium, Discrete Contin. Dyn. Syst., 2013, vol. 33, no. 3, pp. 1033–1047.

    Article  MathSciNet  Google Scholar 

  6. Diacu, F., Relative Equilibria of the \(3\)-Dimensional Curved \(n\)-Body Problem, Mem. Am. Math. Soc., 2014, vol. 228, no. 1071, pp. 86).

    MathSciNet  MATH  Google Scholar 

  7. Diacu, F., Pérez-Chavela, E., and Reyes Victoria, J. G., An Intrinsic Approach in the Curved \(n\)-Body Problem: The Negative Curvature Case, J. Differential Equations, 2012, vol. 252, no. 8, pp. 4529–4562.

    Article  MathSciNet  Google Scholar 

  8. Diacu, F. and Popa, S., All the Lagrangian Relative Equilibria of the Curved \(3\)-Body Problem Have Equal Masses, J. Math. Phys., 2014, vol. 55, no. 11, 112701, 9 pp.

    Article  MathSciNet  Google Scholar 

  9. Diacu, F., Stoica, C., and Zhu, S., Central Configurations of the Curved \(n\)-Body Problem, J. Nonlinear Sci., 2018, vol. 28, no. 5, pp. 1999–2046.

    Article  MathSciNet  Google Scholar 

  10. Diacu, F. and Zhu, S., Almost All \(3\)-Body Relative Equilibria on \(\mathbb{S}^{2}\) and \(\mathbb{H}^{2}\) Are Inclined, Discrete Contin. Dyn. Syst. Ser. S, 2020, vol. 13, no. 4, pp. 1131–1143.

    MathSciNet  MATH  Google Scholar 

  11. García-Naranjo, L. C., Marrero, J. C., Pérez-Chavela, E., and Rodríguez-Olmos, M., Classification and Stability of Relative Equilibria for the Two-Body Problem in the Hyperbolic Space of Dimension \(2\), J. Differential Equations, 2016, vol. 260, no. 7, pp. 6375–6404.

    Article  MathSciNet  Google Scholar 

  12. Kilin, A. A., Libration Points in Spaces \(\mathbf{S}^{2}\) and \(\mathbf{L}^{2}\), Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 91–103.

    Article  MathSciNet  Google Scholar 

  13. Martínez, R. and Simó, C., Relative Equilibria of the Restricted Three-Body Problem in Curved Spaces, Celestial Mech. Dynam. Astronom., 2017, vol. 128, nos. 2–3, pp. 221–259.

    Article  MathSciNet  Google Scholar 

  14. Moeckel, R., Celestial Mechanics: Especially Central Configurations, http://www.math.umn.edu/\~rmoeckel/notes/CMNotes.pdf (2014).

  15. Marsden, J. E., Lectures on Mechanics, ,London Math. Soc. Lecture Note Ser., vol. 174, Cambridge: Cambridge Univ. Press, 1992.

    Google Scholar 

  16. Moulton, F. R., The Straight Line Solutions of the Problem of \(n\) Bodies, Ann. of Math. (2), 1910, vol. 12, no. 1, pp. 1–17.

    Article  MathSciNet  Google Scholar 

  17. Pacella, F., Central Configurations of the \(N\)-Body Problem via Equivariant Morse Theory, Arch. Rational Mech. Anal., 1987, vol. 97, no. 1, pp. 59–74.

    Article  MathSciNet  Google Scholar 

  18. Palmore, J. I., Classifying Relative Equilibria: 1, Bull. Amer. Math. Soc., 1973, vol. 79, pp. 904–908.

    Article  MathSciNet  Google Scholar 

  19. Palmore, J. I., Relative Equilibria of the \(n\)-Body Problem in \(E^{4}\), J. Differential Equations, 1980, vol. 38, no. 2, pp. 278–300.

    Article  MathSciNet  Google Scholar 

  20. Pérez-Chavela, E. and Sánchez-Cerritos, J. M., Hyperbolic Relative Equilibria for the Negative Curved \(n\)-Body Problem, Commun. Nonlinear Sci. Numer. Simul., 2019, vol. 67, pp. 460–479.

    Article  MathSciNet  Google Scholar 

  21. Shub, M., Appendix to Smale’s Paper: “Diagonals and Relative Equilibria,”, in Manifolds: Proc. Nuffic Summer School (Amsterdam, 1970), Lecture Notes in Math., vol. 197, Berlin: Springer, 1971, pp. 199–201.

    Google Scholar 

  22. Smale, S., Topology and Mechanics: 2. The Planar \(n\)-Body Problem, Invent. Math., 1970, vol. 11, no. 1, pp. 45–64.

    Article  MathSciNet  Google Scholar 

  23. Smale, S., Problems on the Nature of Relative Equilibria in Celestial Mechanics, in Manifolds: Proc. Nuffic Summer School (Amsterdam, 1970), Lecture Notes in Math., vol. 197, Berlin: Springer, 1971, pp. 194–198.

    Google Scholar 

  24. Tibboel, P., Existence of a Lower Bound for the Distance between Point Masses of Relative Equilibria in Spaces of Constant Curvature, J. Math. Anal. Appl., 2014, vol. 416, no. 1, pp. 205–211.

    Article  MathSciNet  Google Scholar 

  25. Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton Math. Ser., vol. 5, Princeton, N.J.: Princeton Univ. Press, 1941.

    MATH  Google Scholar 

  26. Yu, X. and Zhu, S., Regular Polygonal Equilibrium Configurations on \(S^{1}\) and Stability of the Associated Relative Equilibria, J. Dyn. Diff. Equ., 2021, vol. 33, no. 2, pp. 1071–1086.

  27. Zhu, S. and Zhao, S., Three-Dimensional Central Configurations in \(\mathbb{H}^{3}\) and \(\mathbb{S}^{3}\), J. Math. Phys., 2017, vol. 58, no. 2, 022901, 7 pp.

    Article  MathSciNet  Google Scholar 

  28. Zhu, S., Dziobek Equilibrium Configurations on a Sphere,https://doi.org/10.1007/s10884-021-10001-9 (2021).

Download references

ACKNOWLEDGMENTS

I would like to thank the referee for many helpful suggestions and pointing out a gap in the original manuscript. The authors thank to Xiang Yu and Ernesto Pérez-Chavela for useful discussions.

Funding

This work is supported by NSFC(No. 11801537) and China Scholarship Council (CSC NO. e201806345013).

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Correspondence to Shuqiang Zhu.

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MSC2010

70F15, 70K42, 34C40

APPENDIX. THE RELATIVE EQUILIBRIA AND CENTRAL CONFIGURATIONS

Recall that \(\mathbb{S}_{xy}^{1}:=\{(x,y,z,w)\in\mathbb{S}^{3}:z=w=0\}\), \(\mathbb{S}_{zw}^{1}:=\{(x,y,z,w)\in\mathbb{S}^{3}:x=y=0\}\). Recall that the critical points of \(U_{1}\) are special central configurations, the critical points of \(U_{1}-\lambda I_{1},U_{-1}-\lambda I_{-1}\) that are not special central configurations are ordinary central configurations. The motions in the form \(\mathbf{q}(t)=\mathbf{q}(0)\) are called equilibria, and the motions in the form \(\exp(t\xi)\mathbf{q}\) (\(\xi\neq 0\)) are called relative equilibria.

Proposition 7 ([9])

  • Let \(\mathbf{q}=(\mathbf{q}_{1},\dots,\mathbf{q}_{n})\in(\mathbb{H}^{3})^{n}\) be an ordinary central configuration with multiplier \(\lambda\) . Then the associated relative equilibria are \(B_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha=\sqrt{-2\lambda}\cos s,\beta=\sqrt{-2\lambda}\sin s,s\in(0,2\pi]\) .

  • Let \(\mathbf{q}=(\mathbf{q}_{1},\dots,\mathbf{q}_{n})\in(\mathbb{S}^{3})^{n}\) be a special central configuration. Then the associated equilibrium is \(\mathbf{q}(t)=\mathbf{q}\) . The associated relative equilibria are

    • \(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha,\beta\in\mathbb{R}\) if all particles are on \(\mathbb{S}^{1}_{xy}\cup\mathbb{S}^{1}_{zw}\) ;

    • \(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\beta=\pm\alpha,\alpha\in\mathbb{R}\) , if not all particles are on \(\mathbb{S}^{1}_{xy}\cup\mathbb{S}^{1}_{zw}\) .

  • Let \(\mathbf{q}=(\mathbf{q}_{1},\dots,\mathbf{q}_{n})\in(\mathbb{S}^{3})^{n}\) be an ordinary central configuration. The associated relative equilibria are

    • \(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha=\sqrt{2\lambda}\sinh s,\beta=\sqrt{2\lambda}\cosh s,\) \(s\in\mathbb{R}\) , if \(\lambda>0\) ;

    • \(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha=\sqrt{-2\lambda}\cosh s,\beta=\sqrt{-2\lambda}\sinh s\) , \(s\in\mathbb{R}\) , if \(\lambda<0\) .

For one ordinary central configuration, among the set of associated relative equilibria, there are periodic ones (in \(H^{3}\), \(\beta=0\); in \(S^{3}\), \(\alpha+k\beta=0\) for some \(k\in\mathbb{Z}\)), and quasi-periodic ones (in \(H^{3}\), none; in \(S^{3}\), \(\alpha+k\beta\neq 0\) for any \(k\in\mathbb{Z}\)).

In the Newtonian \(n\)-body problem, the relative equilibrium is always planar, while in the curved \(n\)-body problem the set of relative equilibria has richer structure. We divide them into three classes: geodesic, 2-dimensional, and 3-dimensional ones. A geodesic relative equilibrium is one with all particles on the same geodesic for all \(t\); a \(2\) -dimensional relative equilibrium is one with all particles on the same 2-dimensional great sphere for all \(t\), but not on the same geodesic; the others are \(3\)-dimensional relative equilibria.

If a \(k\)-dimensional relative equilibrium is associated with an \(m\)-dimensional configuration, then \(k\geqslant m\). Let \(Q(t)\mathbf{q}\) be one relative equilibrium. Then it is a geodesic one if \(\mathbf{q}\) is on a geodesic and \(Q(t)\) keeps that geodesic; it is a 2-dimensional one if \(\mathbf{q}\) is on a 2-dimensional great sphere (\(\mathbf{q}\) may be a geodesic one in that sphere), and \(Q(t)\) keeps that 2-dimensional great sphere.

Let \(G_{1}=SO(2)\times SO^{+}(1,1)\) and \(G_{2}=SO(2)\times SO(2)\). Assume that \(\mathbf{q}\) is an ordinary central configuration in \(\mathbb{H}^{3}\) (resp. \(\mathbb{S}^{3}\)). Then \(g\mathbf{q}\) is an ordinary central configuration with the same multiplier if \(g\) is in \(G_{1}\) (resp. \(G_{2}\)). If \(B_{\alpha,\beta}(t)\mathbf{q}\) (resp. \(A_{\alpha,\beta}(t)\mathbf{q}\)) are relative equilibria associated with \(\mathbf{q}\), then the relative equilibria associated with \(g\mathbf{q}\) are

$$B_{\alpha,\beta}(t)g\mathbf{q}=gB_{\alpha,\beta}(t)\mathbf{q},\ ({\rm resp.}\ A_{\alpha,\beta}(t)g\mathbf{q}=gA_{\alpha,\beta}(t)\mathbf{q}).$$
Let \(\tau\) be the isometry in \(O(4)\), \(\tau(x,y,z,w)=(z,w,x,y)\). By Theorem 5, if \(\mathbf{q}\) is an ordinary central configuration in \(\mathbb{S}^{3}\) with multiplier \(\lambda\), then the multiplier of \(\tau\mathbf{q}\) is \(-\lambda\). If \(A_{\alpha,\beta}(t)\mathbf{q}\) are relative equilibria associated with \(\mathbf{q}\), then the solutions associated with \(\tau\mathbf{q}\) are
$$A_{\beta,\alpha}(t)\tau\mathbf{q}=\tau A_{\alpha,\beta}(t)\mathbf{q}.$$

Thus, thanks to Theorems 5 and 6, to find geodesic and 2-dimensional relative equilibria, it is enough to assume that the associated ordinary central configuration lies on \(\mathbb{H}^{2}_{xyw}\) for the \(\mathbb{H}^{3}\) case, and on \(\mathbb{S}^{2}_{xyz}\) with negative multiplier for the \(\mathbb{S}^{3}\) case.

Proposition 8

Consider the curved \(n\) -body problem in \(\mathbb{H}^{3}\) . Let \(G_{1}=SO(2)\times SO^{+}(1,1)\) .

  • There is no geodesic relative equilibrium.

  • Any \(2\) -dimensional relative equilibria must be in one of the following three forms:

    • \(gB_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) and \(gB_{0,\pm\sqrt{-2\lambda}}(t)\mathbf{q}\) for \(\mathbf{q}\) being a geodesic ordinary central configuration on \(\mathbb{H}^{1}_{xw}\) with multiplier \(\lambda\) ;

    • \(gB_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) for \(\mathbf{q}\) being a 2-dimensional ordinary central configuration on \(\mathbb{H}^{2}_{xyw}\) with multiplier \(\lambda\) ,

    where \(g\) is some isometry in \(G_{1}\) .

  • Any other relative equilibrium is \(3\) -dimensional.

Proof

Let \(\mathbf{q}\) be an ordinary central configuration on \(\mathbb{H}^{1}_{xw}\) with multiplier \(\lambda\). The 1-parameter subgroup \(B_{\alpha,\beta}(t)\) keeps the geodesic \(\mathbb{H}^{1}_{xw}\) only if \(\alpha=\beta=0\), which is impossible since \(\lambda<0\) by Proposition 3 and \(2\lambda=-(\alpha^{2}+\beta^{2})\). So there is no geodesic relative equilibrium. Obviously, the 1-parameter subgroup \(B_{\alpha,\beta}(t)\) keeps a 2-dimensional great sphere containing \(\mathbb{H}^{1}_{xw}\) only if \(\alpha=0\) or \(\beta=0\). If \(\beta=0\) (resp. \(\alpha=0\)), the associated 2-dimensional relative equilibrium is \(B_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) (resp. \(B_{0,\pm\sqrt{-2\lambda}}(t)\mathbf{q}\)).

Let \(\mathbf{q}\) be a 2-dimensional ordinary central configuration on \(\mathbb{H}^{2}_{xyw}\). Obviously, the 1-parameter subgroup \(B_{\alpha,\beta}(t)\) keeps \(\mathbb{H}^{2}_{xyw}\) only if \(\beta=0\). So 2-dimensional relative equilibria associated with \(\mathbf{q}\) are \(B_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\). By the discussion before Proposition 8, the proof is complete.     \(\square\)

Recall that a relative equilibrium \(B_{0,\beta}(t)\mathbf{q}\) is hyperbolic. In [20], Pérez-Chavela and Sánchez-Cerritos consider 2-dimensional hyperbolic relative equilibria. They show that, if the masses are equal, the configuration of such relative equilibria cannot be a regular polygon. In fact, those motions can be characterized as follows:

Proposition 9

All 2-dimensional hyperbolic relative equilibria must be associated with geodesic ordinary central configurations. Given \(n\) masses, there are exactly \(n!\) families of 2-dimensional hyperbolic relative equilibria, one for each ordering of the masses along the geodesic.

Proof

The first part is from the second statement of Proposition 8. We can prove it directly. Since the motion is 2-dimensional, we use the Poincaré half-plane model: \(H,(x,y),y>0,ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}\). Then the kinetics is \(K=\frac{1}{2}\sum m_{i}\frac{\dot{x}_{i}^{2}+\dot{y}_{i}^{2}}{y_{i}^{2}}\). In this model, the hyperbolic 1-parameter subgroup acts on \(H\) by \((x,y)\mapsto e^{\alpha s}(x,y)\) [7]. Thus, the vector field on \(H^{n}\) generated by the hyperbolic 1-parameter subgroup is \(\xi_{H^{n}}(\mathbf{q})=\alpha(x_{1},y_{1},\ldots,x_{n},y_{n})\). Then the augmented potential is

$$U+K(\xi_{H^{n}}(\mathbf{q}))=U+\frac{\alpha^{2}}{2}\sum_{i=1}^{n}m_{i}\frac{x_{i}^{2}}{y_{i}^{2}}+\frac{\alpha^{2}}{2}\sum_{i=1}^{n}m_{i}.$$
If \(\mathbf{q}(t)=e^{\alpha t}\mathbf{q}(0)\) is a hyperbolic relative equilibrium on \(H\), then \(\mathbf{q}(0)\) is a critical point of the above augmented potential. That is, \(\mathbf{q}(0)\) must satisfy the equation
$$\nabla_{\mathbf{q}_{i}}U=-\frac{\alpha^{2}}{2}\nabla_{\mathbf{q}_{i}}\sum_{i=1}^{n}m_{i}\frac{x_{i}^{2}}{y_{i}^{2}}=-\alpha^{2}m_{i}\frac{1}{y_{i}}(x_{i}y_{i},-x_{i}^{2}).$$
(A.1)
We claim that the critical points of this potential must be geodesic configurations. Recall that the geodesics on \(H\) are straight lines and circles perpendicular to the \(x\)-axis. Assume that the particles of \(\mathbf{q}(0)\) are distributed on several circular geodesics \(x^{2}+y^{2}=R_{j}^{2},j=1,\ldots,p\) and that \(R_{j}\leqslant R_{p}\) if \(1\leqslant j\leqslant p\). Consider Eq. (A.1) for one particle, say \(\mathbf{q}_{1}\), on the largest circle. Note that the right-hand side of (A.1) is a vector tangent to the largest circle, but the left-hand side, the force exerted on \(\mathbf{q}_{1}\), is pointing inwards since there are particles on some smaller circle. This contradiction shows that the critical points of the augmented potential have to be geodesic configurations.

The second part is from Theorem 7.     \(\square\)

Note that, for all 2-dimensional hyperbolic relative equilibria, the velocities are orthogonal to the geodesic containing the configuration. This is true for any relative equilibria associated to a geodesic configuration. By Proposition 8, we may assume that the geodesic is \(\mathbb{H}^{1}_{xw}\). Then the velocity of the \(i\)th particle is \((0,\alpha x_{i},\beta w_{i},0)\) for some \(\alpha,\beta\), which is orthogonal to the geodesic.

Proposition 10

For the curved \(n\) -body problem in \(\mathbb{S}^{3}\) , consider the relative equilibria associated with ordinary central configurations. Let \(\tau\) be the isometry \(\tau(x,y,z,w)=(z,w,x,y)\) and \(G_{2}=SO(2)\times SO(2)\) .

  • There is no geodesic relative equilibrium.

  • Any \(2\) -dimensional relative equilibria must be in one of the following four forms:

    • \(gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) and \(\tau gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) for \(\mathbf{q}\) being a geodesic ordinary central configuration on \(\mathbb{S}^{1}_{xz}\) with multiplier \(\lambda<0\) .

    • \(gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) and \(\tau gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) for \(\mathbf{q}\) being a \(2\) -dimensional ordinary central configuration on \(\mathbb{S}^{2}_{xyz}\) with multiplier \(\lambda<0\) ,

    where \(g\) is some isometry in \(G_{2}\) .

  • Any other relative equilibrium is \(3\) -dimensional.

We omit the proof since it is similar to that of Proposition 8. Note that the ordinary central configurations on \(\mathbb{S}^{2}_{xyz}\) with multiplier \(\lambda>0\) lead to 3-dimensional relative equilibria. Similar to the hyperbolic case, the velocity of any relative equilibrium associated to a geodesic configuration is orthogonal to the geodesic containing the configuration.

We now turn to the special central configurations. In this case, the velocity of relative equilibrium associated to a geodesic configuration may or may not be along the geodesic containing the configuration. There is no need to discuss the dimension of equilibrium solutions.

Proposition 11

For the curved \(n\) -body problem in \(\mathbb{S}^{3}\) , consider the relative equilibria associated with special central configurations.

  • Any geodesic relative equilibrium must be \(A_{\alpha,0}(t)\mathbf{q}\) , or \(\tau A_{\alpha,0}(t)\mathbf{q},\alpha\neq\mathbb{R}\) for \(\mathbf{q}\) being a special central configuration on \(\mathbb{S}^{1}_{xy}\) .

  • There is no \(2\) -dimensional relative equilibrium.

  • Any other relative equilibrium is \(3\) -dimensional.

Proof

The symmetry group for special central configurations is \(O(4)\). Any geodesic special central configuration is in the form \(g\mathbf{q}\), where \(\mathbf{q}\) is on \(\mathbb{S}^{1}_{xy}\) and \(g\in O(4)\). If \(g\mathbf{q}\) is on \(\mathbb{S}^{1}_{xy}\) (resp. \(\mathbb{S}^{1}_{zw}\)), by Proposition 7, the associated relative equilibria are \(A_{\alpha,\beta}(t)\mathbf{q}\) for any \(\alpha,\beta\in\mathbb{R}\), which are the same as \(A_{\alpha,0}(t)\mathbf{q}\) (resp. \(\tau A_{\alpha,0}(t)\mathbf{q}\)). If \(g\mathbf{q}\) is not on \(\mathbb{S}^{1}_{xy}\) nor on \(\mathbb{S}^{1}_{zw}\), by Proposition 7, the associated relative equilibria are \(A_{\alpha,\pm\alpha}(t)\mathbf{q}\), \(\alpha\in\mathbb{R}\). The relative equilibrium is geodesic only if \(\alpha=0\), and it is not geodesic nor 2-dimensional otherwise.

Let \(\mathbf{q}\) be a 2-dimensional special central configuration. We claim that \(\mathbf{q}\) cannot be within \(\mathbb{S}^{1}_{xy}\bigcup\mathbb{S}^{1}_{zw}\). Note that the particles on one of the two circles must be collinear since the configuration is contained in a 3-dimensional hyperplane. Assume that the particles on \(\mathbb{S}^{1}_{xy}\) are collinear. The number of particles on \(\mathbb{S}^{1}_{xy}\) is one, otherwise \(\mathbf{q}\in\Delta_{+}\). Then all particles of \(\mathbf{q}\) are within one 2-dimensional hemisphere, which is impossible, see Section 12.3 of [6] or [28]. The contradiction proves the claim. Hence, the associated relative equilibria must be \(A_{\alpha,\pm\alpha}(t)\mathbf{q}\), \(\alpha\in\mathbb{R}\). The relative equilibrium is 2-dimensional only if \(\alpha=0\).     \(\square\)

To study the 2-dimensional relative equilibria of the curved \(n\)-body problem, instead of restricting to a 2-dimensional physical space directly, i. e., to \(T((\mathbb{S}^{2})^{n}-\Delta_{+})\) or \(T((\mathbb{H}^{2})^{n}-\Delta_{-})\), it seems convenient to start with the configurations and to use the augmented potentials introduced in Theorem 1 in some proper coordinates. Moreover, this restriction would make counting the configurations clumsy.

Let us finish the discussion with one concrete example. In [8], Diacu and Sergiu consider 3-dimensional relative equilibria of three bodies in \(\mathbb{S}^{3}\) with the following property: the configuration is not geodesic and the three mutual distances are the same. They show that the three masses must be equal. This result can be obtained quickly as follows:

The associated central configuration must be 2-dimensional since there are only three bodies and the configuration is not geodesic. It cannot be a special central configuration since it is not geodesic [28]. We may assume that it is on \(\mathbb{S}^{2}_{xyz}\), i. e., \(\mathbf{q}_{i}=(x_{i},y_{i},z_{i})\). According to Diacu and Zhu [10], a three-body configuration on \(\mathbb{S}^{2}_{xyz}\) is an ordinary central configuration if and only if

$$\sum m_{i}z_{i}x_{i}=\sum m_{i}z_{i}y_{i}=0,\ {\rm}\ (z_{1},z_{2},z_{3})=k(\sin^{3}d_{23},\sin^{3}d_{13},\sin^{3}d_{12}).$$
Since the three mutual distances are the same, we find immediately that the three masses are the same.

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Zhu, S. Compactness and Index of Ordinary Central Configurations for the Curved \(N\)-Body Problem. Regul. Chaot. Dyn. 26, 236–257 (2021). https://doi.org/10.1134/S1560354721030035

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