September  2021, 20(9): 2941-2963. doi: 10.3934/cpaa.2021090

The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces

1. 

Department of Mathematics, Shaqra University, Saudi Arabia

2. 

Department of Mathematics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

3. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China

4. 

Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing, China

* Corresponding author

Received  May 2020 Revised  April 2021 Published  September 2021 Early access  June 2021

Fund Project: The second author has been partially supported by Conacyt-México project A1-S-10112 and Asociación Mexicana de Cultura A.C.

We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature $ \kappa $, which without loss of generality we assume $ \kappa = -1 $. Using this model, we first derive the equations of motion for the $ 2 $-and $ 3 $-center problems. We prove that for $ 2 $–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant $ y $–axis, we prove that it is a center, but for the general two center problem it is unstable. For the $ 3 $–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the $ y $–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation.

Citation: Sawsan Alhowaity, Ernesto Pérez-Chavela, Juan Manuel Sánchez-Cerritos. The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces. Communications on Pure & Applied Analysis, 2021, 20 (9) : 2941-2963. doi: 10.3934/cpaa.2021090
References:
[1]

S. AlhowaityF. Diacu and E. Pérez-Chavela, Relative equilibria in curved restricted 4-body problems, Can. Math. Bull., 61 (2018), 673-687.  doi: 10.4153/CMB-2018-019-9.  Google Scholar

[2]

J. AndradeN. DávilaE. Pérez-Chavela and C. Vidal, Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature, Can. J. Math., 69 (2017), 961-991.  doi: 10.4153/CJM-2016-014-5.  Google Scholar

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S. V. Bolotin and P. Negrini, Chaotic behavior in the 3-center problem, J. Differ. Equ., 190 (2003), 539-558. doi: 10.1016/S0022-0396(03)00024-X.  Google Scholar

[4]

W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Teubner, Leipzig-Berlin, 1913. Google Scholar

[5]

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

[6] F. Diacu, Relative equilibria of the curved $N$-body problem, in Atlantis Studies in Dynamical Systems, vol. 1, Atlantis Press, Amsterdam, 2012.  doi: 10.2991/978-94-91216-68-8.  Google Scholar
[7]

F. Diacu, Relative equilibria of the 3-dimensional curved $n$-body problem, Memoirs Amer. Math. Soc., 228 (2013), 1071.  Google Scholar

[8]

F. Diacu, The curved $N$-body problem: risks and rewards, Math. Intell., 35 (2013), 24-33.  doi: 10.1007/s00283-013-9397-1.  Google Scholar

[9]

F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differ. Equ., 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009.  Google Scholar

[10]

F. DiacuR. MartínezE. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Physica D, 256 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007.  Google Scholar

[11]

F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved $3$-body problem, J. Differ. Equ., 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011.  Google Scholar

[12]

F. DiacuE. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear $N$-body problem, Trans. Amer. Math. Soc., 357 (2005), 4215-4223.  doi: 10.1090/S0002-9947-04-03606-2.  Google Scholar

[13]

F. DiacuE. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part I: Relative equilibria, J. Nonlinear Sci., 22 (2012), 247-266.  doi: 10.1007/s00332-011-9116-z.  Google Scholar

[14]

F. DiacuE. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part II: Singularities, J. Nonlinear Sci., 22 (2012), 267-275.  doi: 10.1007/s00332-011-9117-y.  Google Scholar

[15]

F. DiacuE. Pérez-Chavela and G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The negative curvature case, J. Differ. Equ., 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.  Google Scholar

[16]

F. Diacu and §. Popa, All Lagrangian relative equilibria have equal masses, J. Math. Phys., 55 (2014), 112701. doi: 10.1063/1.4900833.  Google Scholar

[17]

F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc., 143 (2015), 1583-1593.  doi: 10.1090/S0002-9939-2014-12326-4.  Google Scholar

[18]

L. C. García-NaranjoJ. C. MarreroE. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differ. Equ., 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.  Google Scholar

[19]

W. Killing, Die Rechnung in den nichteuklidischen Raumformen, J. Reine Angew. Math., 89 (1880), 265-287.  doi: 10.1515/crll.1880.89.265.  Google Scholar

[20]

W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math., 98 (1885), 1-48.  doi: 10.1515/crll.1885.98.1.  Google Scholar

[21]

W. Killing, Die Nicht-Euklidischen Raumformen in Analytischer Behandlung, Teubner, Leipzig, 1885. Google Scholar

[22]

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

[23]

H. Kragh, Is space Flat? Nineteenth century astronomy and non-Euclidean geometry, J. Astr. Hist. Heritage, 15 (2012), 149-158.   Google Scholar

[24]

R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12 (1873), 349-370.   Google Scholar

[25]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, in Collected Works, vol. 2, GITTL, Moscow, 1949. Google Scholar

[26]

R. Martínez and C. Simó, On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb S^2$, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013) 1157–1175. doi: 10.3934/dcds. 2013.33.1157.  Google Scholar

[27]

R. Martínez and C. Simó, Relative equilibria of the restricted 3-body problem in curved spaces, Celes.t Mech. Dyn. Ast.r, 128 (2017), 221-259.  doi: 10.1007/s10569-016-9750-8.  Google Scholar

[28]

E. Pérez-Chavela and J. G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The positive curvature case, Trans. Amer. Math. Soc., 364 (2012), 3805-3827.  doi: 10.1090/S0002-9947-2012-05563-2.  Google Scholar

[29]

E. Schering, Die Schwerkraft im Gaussischen Räume, Nachr. Königl. Ges. Wiss. Gött., 15, (1870), 311–321. Google Scholar

[30]

E. Schering, Die Schwerkraft in mehrfach ausgedehnten Gaussischen und Riemmanschen Räumen, Nachr. Königl. Ges. Wiss. Gött., 6 (1873), 149–159. Google Scholar

[31]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A: Math. Gen., 39, (2006), 5787-5806; corrected version at math. DS/0601382. doi: 10.1088/0305-4470/39/20/011.  Google Scholar

[32]

P. Tibboel, Polygonal homographic orbits in spaces of constant curvature, Proc. Amer. Math. Soc., 141 (2013), 1465-1471.  doi: 10.1090/S0002-9939-2012-11410-8.  Google Scholar

[33]

P. Tibboel, Existence of a class of rotopulsators, J. Math. Anal. Appl., 404 (2013), 185-191.  doi: 10.1016/j.jmaa.2013.02.066.  Google Scholar

[34]

P. Tibboel, Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature, J. Math. Anal. Appl., 416 (2014), 205-211.  doi: 10.1016/j.jmaa.2014.02.036.  Google Scholar

[35]

T. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, in Astrophysics and Space Science Library, Volume 295, Springer, 2003. doi: 10.1007/978-94-017-0303-1.  Google Scholar

[36]

S. Zhu, Eulerian relative equilibria of the curved 3-body problems in $\mathbb S^2$, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2.  Google Scholar

show all references

References:
[1]

S. AlhowaityF. Diacu and E. Pérez-Chavela, Relative equilibria in curved restricted 4-body problems, Can. Math. Bull., 61 (2018), 673-687.  doi: 10.4153/CMB-2018-019-9.  Google Scholar

[2]

J. AndradeN. DávilaE. Pérez-Chavela and C. Vidal, Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature, Can. J. Math., 69 (2017), 961-991.  doi: 10.4153/CJM-2016-014-5.  Google Scholar

[3]

S. V. Bolotin and P. Negrini, Chaotic behavior in the 3-center problem, J. Differ. Equ., 190 (2003), 539-558. doi: 10.1016/S0022-0396(03)00024-X.  Google Scholar

[4]

W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Teubner, Leipzig-Berlin, 1913. Google Scholar

[5]

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

[6] F. Diacu, Relative equilibria of the curved $N$-body problem, in Atlantis Studies in Dynamical Systems, vol. 1, Atlantis Press, Amsterdam, 2012.  doi: 10.2991/978-94-91216-68-8.  Google Scholar
[7]

F. Diacu, Relative equilibria of the 3-dimensional curved $n$-body problem, Memoirs Amer. Math. Soc., 228 (2013), 1071.  Google Scholar

[8]

F. Diacu, The curved $N$-body problem: risks and rewards, Math. Intell., 35 (2013), 24-33.  doi: 10.1007/s00283-013-9397-1.  Google Scholar

[9]

F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differ. Equ., 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009.  Google Scholar

[10]

F. DiacuR. MartínezE. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Physica D, 256 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007.  Google Scholar

[11]

F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved $3$-body problem, J. Differ. Equ., 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011.  Google Scholar

[12]

F. DiacuE. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear $N$-body problem, Trans. Amer. Math. Soc., 357 (2005), 4215-4223.  doi: 10.1090/S0002-9947-04-03606-2.  Google Scholar

[13]

F. DiacuE. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part I: Relative equilibria, J. Nonlinear Sci., 22 (2012), 247-266.  doi: 10.1007/s00332-011-9116-z.  Google Scholar

[14]

F. DiacuE. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part II: Singularities, J. Nonlinear Sci., 22 (2012), 267-275.  doi: 10.1007/s00332-011-9117-y.  Google Scholar

[15]

F. DiacuE. Pérez-Chavela and G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The negative curvature case, J. Differ. Equ., 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.  Google Scholar

[16]

F. Diacu and §. Popa, All Lagrangian relative equilibria have equal masses, J. Math. Phys., 55 (2014), 112701. doi: 10.1063/1.4900833.  Google Scholar

[17]

F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc., 143 (2015), 1583-1593.  doi: 10.1090/S0002-9939-2014-12326-4.  Google Scholar

[18]

L. C. García-NaranjoJ. C. MarreroE. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differ. Equ., 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.  Google Scholar

[19]

W. Killing, Die Rechnung in den nichteuklidischen Raumformen, J. Reine Angew. Math., 89 (1880), 265-287.  doi: 10.1515/crll.1880.89.265.  Google Scholar

[20]

W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math., 98 (1885), 1-48.  doi: 10.1515/crll.1885.98.1.  Google Scholar

[21]

W. Killing, Die Nicht-Euklidischen Raumformen in Analytischer Behandlung, Teubner, Leipzig, 1885. Google Scholar

[22]

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

[23]

H. Kragh, Is space Flat? Nineteenth century astronomy and non-Euclidean geometry, J. Astr. Hist. Heritage, 15 (2012), 149-158.   Google Scholar

[24]

R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12 (1873), 349-370.   Google Scholar

[25]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, in Collected Works, vol. 2, GITTL, Moscow, 1949. Google Scholar

[26]

R. Martínez and C. Simó, On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb S^2$, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013) 1157–1175. doi: 10.3934/dcds. 2013.33.1157.  Google Scholar

[27]

R. Martínez and C. Simó, Relative equilibria of the restricted 3-body problem in curved spaces, Celes.t Mech. Dyn. Ast.r, 128 (2017), 221-259.  doi: 10.1007/s10569-016-9750-8.  Google Scholar

[28]

E. Pérez-Chavela and J. G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The positive curvature case, Trans. Amer. Math. Soc., 364 (2012), 3805-3827.  doi: 10.1090/S0002-9947-2012-05563-2.  Google Scholar

[29]

E. Schering, Die Schwerkraft im Gaussischen Räume, Nachr. Königl. Ges. Wiss. Gött., 15, (1870), 311–321. Google Scholar

[30]

E. Schering, Die Schwerkraft in mehrfach ausgedehnten Gaussischen und Riemmanschen Räumen, Nachr. Königl. Ges. Wiss. Gött., 6 (1873), 149–159. Google Scholar

[31]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A: Math. Gen., 39, (2006), 5787-5806; corrected version at math. DS/0601382. doi: 10.1088/0305-4470/39/20/011.  Google Scholar

[32]

P. Tibboel, Polygonal homographic orbits in spaces of constant curvature, Proc. Amer. Math. Soc., 141 (2013), 1465-1471.  doi: 10.1090/S0002-9939-2012-11410-8.  Google Scholar

[33]

P. Tibboel, Existence of a class of rotopulsators, J. Math. Anal. Appl., 404 (2013), 185-191.  doi: 10.1016/j.jmaa.2013.02.066.  Google Scholar

[34]

P. Tibboel, Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature, J. Math. Anal. Appl., 416 (2014), 205-211.  doi: 10.1016/j.jmaa.2014.02.036.  Google Scholar

[35]

T. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, in Astrophysics and Space Science Library, Volume 295, Springer, 2003. doi: 10.1007/978-94-017-0303-1.  Google Scholar

[36]

S. Zhu, Eulerian relative equilibria of the curved 3-body problems in $\mathbb S^2$, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2.  Google Scholar

Figure 1.  Phase space for $ y(t),\dot{y}(t) $ ($ x(t) = 0 $)
Figure 2.  The vertical case
Figure 3.  Phase portrait in non-regularized and regularized coordinates
Figure 4.  The geodesic case
Figure 5.  The geodesic case, with $ M $ between $ \mu $ and $ m_1 $
Figure 6.  Phase portrait in n non-regularized and regularized coordinates
Figure 7.  Shaped area corresponds to the points $ ( \xi, \eta) $ with $ \cos \xi \cosh \eta<\frac{B}{A} $ with $ \bar{y}_1 = 2 $. Region 1 is for $ x>0 $, $ y>0 $; and region 2 for $ x<0 $, $ y>0 $
Figure 8.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure 9.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure 10.  We plot the same constrained graphic as Figure 9b where we can see the positions of the centers plotted
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