# American Institute of Mathematical Sciences

April  2020, 13(4): 1131-1143. doi: 10.3934/dcdss.2020067

## Almost all 3-body relative equilibria on $\mathbb S^2$ and $\mathbb H^2$ are inclined

 1 Yale-NUS College, National University of Singapore, Republic of Singapore 2 School of Mathematical Sciences, University of Science and Technology of China, Hefei, China

* Corresponding author: Shuqiang Zhu

Dedicated to Jürgen Scheurle on the occasion of his 65th birthday
Editors' Note: Florin Diacu passed away on February 13, 2018 before this manuscript could be published. He will be missed by his colleagues, as a mathematician and as a person.

Received  November 2017 Revised  March 2018 Published  April 2019

Fund Project: Florin Diacu is supported by Yale-NUS startup grant, and Shuqiang Zhu is supported by NSFC(No.11801537, No.11721101) and the Fundamental Research Funds for the Central Universities (No.WK0010450010).

We answer here a question posed by F. Diacu in 2012 that asked whether there exist relative equilibria on $\mathbb S^2$ and $\mathbb H^2$ that move in a plane non-perpendicular to the rotation axis. For 3-body non-geodesic ordinary central configurations on $\mathbb S^2$ and $\mathbb H^2$, we find all relative equilibria that move in a plane perpendicular to the rotation axis. We also show that the set of shapes of 3-body non-geodesic ordinary central configurations on $\mathbb S^2$ and $\mathbb H^2$ is a 3-dimensional manifold. Then we conclude that almost all 3-body relative equilibria move in planes non-perpendicular to the rotation axis.

Citation: Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $\mathbb S^2$ and $\mathbb H^2$ are inclined. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067
##### References:
 [1] F. Diacu, On the singularities of the curved $n$-body problem, Trans. Amer. Math. Soc., 363 (2011), 2249-2264.  doi: 10.1090/S0002-9947-2010-05251-1. [2] F. Diacu, Polygonal homographic orbits of the curved $n$-body problem, Trans. Amer. Math. Soc., 364 (2012), 2783-2802.  doi: 10.1090/S0002-9947-2011-05558-3. [3] F. Diacu, Relative Equilibria of the Curved N-Body Problem, Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012.  doi: 10.2991/978-94-91216-68-8. [4] F. Diacu, Relative equilibria in the 3-dimensional curved $n$-body problem, Mem. Amer. Math. Soc., 228 (2014), ⅵ+80 pp. [5] F. Diacu, Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem, J. Math. Phys., 57 (2016), 112701, 20pp. doi: 10.1063/1.4967443. [6] F. Diacu, The classical N-body problem in the context of curved space, Canad. J. Math., 69 (2017), 790-806.  doi: 10.4153/CJM-2016-041-2. [7] F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differential Equations, 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009. [8] F. Diacu, R. Martínez, E. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Phys. D, 256/257 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007. [9] F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved 3-body problem, J. Differential Equations, 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011. [10] F. Diacu and S. Popa, All the Lagrangian relative equilibria of the curved 3-body problem have equal masses, J. Math. Phys., 55 (2014), 112701, 9pp. doi: 10.1063/1.4900833. [11] F. Diacu, J. M. Sánchez-Cerritos and S. Zhu, Stability of fixed Points and associated relative equilibria of the 3-body problem on ${\Bbb {S}}^1$ and ${\Bbb {S}}^2$, J. Dynam. Differential Equations, 30 (2018), 209-225.  doi: 10.1007/s10884-016-9550-6. [12] F. Diacu, C. Stoica and S. Zhu, Central configurations of the curved N-body problem, J. Nonlinear Sci., 28 (2018), 1999–2046, arXiv: 1603.03342. doi: 10.1007/s00332-018-9473-y. [13] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. [14] A. A. Kilin, Libration points in spaces S2 and L2, Regul. Chaotic Dyn., 4 (1999), 91-103.  doi: 10.1070/rd1999v004n01ABEH000101. [15] V. V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 54 (1992), 393-399.  doi: 10.1007/BF00049149. [16] J. Llibre, R. Moeckel and C. Simó, Central Configurations, Periodic Orbits, and Hamiltonian Systems, Advanced Courses in Mathematics. CRM Barcelona, Lecture notes given at the Centre de Recerca Matemàtica (CRM), Barcelona, January 27–31, 2014, Edited by Montserrat Corbera, Josep Maria Cors and Enrique Ponce, Birkhäuser Springer, Basel, 2015. doi: 10.1007/978-3-0348-0933-7. [17] D. G. Saari, On the role and the properties of n-body central configurations, Celestial Mech., 21 (1980), 9-20.  doi: 10.1007/BF01230241. [18] E. Schrödinger, A method for determining quantum-mechanical eigenvalues and eigenfunctions, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 46 (1940), 9-16. [19] A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A, 39 (2006), 5787-5806. [20] A. V. Shchepetilov, Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces, Lecture Notes in Physics, Springer, Berlin, 2006. [21] S. Smale, Topology and mechanics. Ⅱ. The planar n-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805. [22] S. Smale, Problems on the nature of relative equilibria in celestial mechanics, in Manifolds –Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, (1971), 194–198. [23] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, ⅴ. 5, Princeton University Press, Princeton, N. J., 1941. [24] S. Zhu, Eulerian relative equilibria of the curved 3-body problems in S2, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2. [25] S. Zhu and S. Zhao, Three-dimensional central configurations in ${\Bbb {H}}^3$ and ${\Bbb {S}}^3$, J. Math. Phys., 58 (2017), 022901, 7pp. doi: 10.1063/1.4975214. [26] S. Zhu, A lower bound for the number of central configurations on ${\mathbb {H}}^2$, preprint, arXiv: 1702.05535. [27] S. Zhu, On Dziobek special central configurations, preprint, arXiv: 1705.03987.

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##### References:
 [1] F. Diacu, On the singularities of the curved $n$-body problem, Trans. Amer. Math. Soc., 363 (2011), 2249-2264.  doi: 10.1090/S0002-9947-2010-05251-1. [2] F. Diacu, Polygonal homographic orbits of the curved $n$-body problem, Trans. Amer. Math. Soc., 364 (2012), 2783-2802.  doi: 10.1090/S0002-9947-2011-05558-3. [3] F. Diacu, Relative Equilibria of the Curved N-Body Problem, Atlantis Studies in Dynamical Systems, Atlantis Press, Paris, 2012.  doi: 10.2991/978-94-91216-68-8. [4] F. Diacu, Relative equilibria in the 3-dimensional curved $n$-body problem, Mem. Amer. Math. Soc., 228 (2014), ⅵ+80 pp. [5] F. Diacu, Bifurcations of the Lagrangian orbits from the classical to the curved 3-body problem, J. Math. Phys., 57 (2016), 112701, 20pp. doi: 10.1063/1.4967443. [6] F. Diacu, The classical N-body problem in the context of curved space, Canad. J. Math., 69 (2017), 790-806.  doi: 10.4153/CJM-2016-041-2. [7] F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differential Equations, 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009. [8] F. Diacu, R. Martínez, E. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Phys. D, 256/257 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007. [9] F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved 3-body problem, J. Differential Equations, 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011. [10] F. Diacu and S. Popa, All the Lagrangian relative equilibria of the curved 3-body problem have equal masses, J. Math. Phys., 55 (2014), 112701, 9pp. doi: 10.1063/1.4900833. [11] F. Diacu, J. M. Sánchez-Cerritos and S. Zhu, Stability of fixed Points and associated relative equilibria of the 3-body problem on ${\Bbb {S}}^1$ and ${\Bbb {S}}^2$, J. Dynam. Differential Equations, 30 (2018), 209-225.  doi: 10.1007/s10884-016-9550-6. [12] F. Diacu, C. Stoica and S. Zhu, Central configurations of the curved N-body problem, J. Nonlinear Sci., 28 (2018), 1999–2046, arXiv: 1603.03342. doi: 10.1007/s00332-018-9473-y. [13] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. [14] A. A. Kilin, Libration points in spaces S2 and L2, Regul. Chaotic Dyn., 4 (1999), 91-103.  doi: 10.1070/rd1999v004n01ABEH000101. [15] V. V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 54 (1992), 393-399.  doi: 10.1007/BF00049149. [16] J. Llibre, R. Moeckel and C. Simó, Central Configurations, Periodic Orbits, and Hamiltonian Systems, Advanced Courses in Mathematics. CRM Barcelona, Lecture notes given at the Centre de Recerca Matemàtica (CRM), Barcelona, January 27–31, 2014, Edited by Montserrat Corbera, Josep Maria Cors and Enrique Ponce, Birkhäuser Springer, Basel, 2015. doi: 10.1007/978-3-0348-0933-7. [17] D. G. Saari, On the role and the properties of n-body central configurations, Celestial Mech., 21 (1980), 9-20.  doi: 10.1007/BF01230241. [18] E. Schrödinger, A method for determining quantum-mechanical eigenvalues and eigenfunctions, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 46 (1940), 9-16. [19] A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A, 39 (2006), 5787-5806. [20] A. V. Shchepetilov, Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces, Lecture Notes in Physics, Springer, Berlin, 2006. [21] S. Smale, Topology and mechanics. Ⅱ. The planar n-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805. [22] S. Smale, Problems on the nature of relative equilibria in celestial mechanics, in Manifolds –Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, (1971), 194–198. [23] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, ⅴ. 5, Princeton University Press, Princeton, N. J., 1941. [24] S. Zhu, Eulerian relative equilibria of the curved 3-body problems in S2, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2. [25] S. Zhu and S. Zhao, Three-dimensional central configurations in ${\Bbb {H}}^3$ and ${\Bbb {S}}^3$, J. Math. Phys., 58 (2017), 022901, 7pp. doi: 10.1063/1.4975214. [26] S. Zhu, A lower bound for the number of central configurations on ${\mathbb {H}}^2$, preprint, arXiv: 1702.05535. [27] S. Zhu, On Dziobek special central configurations, preprint, arXiv: 1705.03987.
Lagrangian central configurations on $\mathbb H^2$
An $\mathbb S^2$ central configuration on $z = c$
The projection of one shape
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