American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020067

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

Florin Diacu 1,†, and  Shuqiang Zhu 2,,

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.

Keywords: Celestial mechanics, curved 3-body problem, relative equilibria, central configurations, three-body problem.
Mathematics Subject Classification: Primary: 70F15; Secondary: 70F07.
Citation: Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[4]

F. Diacu, Relative equilibria in the 3-dimensional curved $n$-body problem, Mem. Amer. Math. Soc., 228 (2014), ⅵ+80 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[14]

A. A. Kilin, Libration points in spaces S2 and L2, Regul. Chaotic Dyn., 4 (1999), 91-103.  doi: 10.1070/rd1999v004n01ABEH000101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A, 39 (2006), 5787-5806.   Google Scholar

[20]

A. V. Shchepetilov, Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces, Lecture Notes in Physics, Springer, Berlin, 2006.  Google Scholar

[21]

S. Smale, Topology and mechanics. Ⅱ. The planar n-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805.  Google Scholar

[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.  Google Scholar

[23] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, ⅴ. 5, Princeton University Press, Princeton, N. J., 1941.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[26]

S. Zhu, A lower bound for the number of central configurations on $ {\mathbb {H}}^2 $, preprint, arXiv: 1702.05535. Google Scholar

[27]

S. Zhu, On Dziobek special central configurations, preprint, arXiv: 1705.03987. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[4]

F. Diacu, Relative equilibria in the 3-dimensional curved $n$-body problem, Mem. Amer. Math. Soc., 228 (2014), ⅵ+80 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[14]

A. A. Kilin, Libration points in spaces S2 and L2, Regul. Chaotic Dyn., 4 (1999), 91-103.  doi: 10.1070/rd1999v004n01ABEH000101.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A, 39 (2006), 5787-5806.   Google Scholar

[20]

A. V. Shchepetilov, Calculus and Mechanics on Two-Point Homogeneous Riemannian Spaces, Lecture Notes in Physics, Springer, Berlin, 2006.  Google Scholar

[21]

S. Smale, Topology and mechanics. Ⅱ. The planar n-body problem, Invent. Math., 11 (1970), 45-64.  doi: 10.1007/BF01389805.  Google Scholar

[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.  Google Scholar

[23] A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton Mathematical Series, ⅴ. 5, Princeton University Press, Princeton, N. J., 1941.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[26]

S. Zhu, A lower bound for the number of central configurations on $ {\mathbb {H}}^2 $, preprint, arXiv: 1702.05535. Google Scholar

[27]

S. Zhu, On Dziobek special central configurations, preprint, arXiv: 1705.03987. Google Scholar

Figure 1.  Lagrangian central configurations on $ \mathbb H^2 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  An $ \mathbb S^2 $ central configuration on $ z = c $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The projection of one shape
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alain Chenciner, Jacques Féjoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 421-438. doi: 10.3934/dcdsb.2008.10.421
[2]

Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157
[3]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[4]

Edward Belbruno. Random walk in the three-body problem and applications. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 519-540. doi: 10.3934/dcdss.2008.1.519
[5]

Martha Alvarez, Joaquin Delgado, Jaume Llibre. On the spatial central configurations of the 5--body problem and their bifurcations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 505-518. doi: 10.3934/dcdss.2008.1.505
[6]

Eduardo Piña. Computing collinear 4-Body Problem central configurations with given masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1215-1230. doi: 10.3934/dcds.2013.33.1215
[7]

Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745
[8]

Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35
[9]

Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609
[10]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[11]

Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631
[12]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[13]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158
[14]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[15]

Montserrat Corbera, Jaume Llibre. On the existence of bi--pyramidal central configurations of the $n+2$--body problem with an $n$--gon base. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1049-1060. doi: 10.3934/dcds.2013.33.1049
[16]

Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323
[17]

Florian Rupp, Jürgen Scheurle. Classification of a class of relative equilibria in three body coulomb systems. Conference Publications, 2011, 2011 (Special) : 1254-1262. doi: 10.3934/proc.2011.2011.1254
[18]

Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090
[19]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062
[20]

Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (14)
  • HTML views (400)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences