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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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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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])
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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 \(\alpha=\sqrt{-2\lambda}\cos s,\beta=\sqrt{-2\lambda}\sin s,s\in(0,2\pi]\) .
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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
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\(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha,\beta\in\mathbb{R}\) if all particles are on \(\mathbb{S}^{1}_{xy}\cup\mathbb{S}^{1}_{zw}\) ;
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\(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\beta=\pm\alpha,\alpha\in\mathbb{R}\) , if not all particles are on \(\mathbb{S}^{1}_{xy}\cup\mathbb{S}^{1}_{zw}\) .
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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
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\(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha=\sqrt{2\lambda}\sinh s,\beta=\sqrt{2\lambda}\cosh s,\) \(s\in\mathbb{R}\) , if \(\lambda>0\) ;
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\(A_{\alpha,\beta}(t)\mathbf{q}\) with \(\alpha=\sqrt{-2\lambda}\cosh s,\beta=\sqrt{-2\lambda}\sinh s\) , \(s\in\mathbb{R}\) , if \(\lambda<0\) .
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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
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 \(G_{1}=SO(2)\times SO^{+}(1,1)\) .
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There is no geodesic relative equilibrium.
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Any \(2\) -dimensional relative equilibria must be in one of the following three forms:
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\(gB_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) 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 \(\lambda\) ;
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\(gB_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) for \(\mathbf{q}\) being a 2-dimensional ordinary central configuration on \(\mathbb{H}^{2}_{xyw}\) with multiplier \(\lambda\) ,
where \(g\) is some isometry in \(G_{1}\) .
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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
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 \(G_{2}=SO(2)\times SO(2)\) .
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There is no geodesic relative equilibrium.
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Any \(2\) -dimensional relative equilibria must be in one of the following four forms:
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\(gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) 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 \(\lambda<0\) .
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\(gA_{\pm\sqrt{-2\lambda},0}(t)\mathbf{q}\) 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 \(G_{2}\) .
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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.
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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 \(\mathbb{S}^{1}_{xy}\) .
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There is no \(2\) -dimensional relative equilibrium.
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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
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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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DOI: https://doi.org/10.1134/S1560354721030035