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

Sawsan Alhowaity; Ernesto Pérez-Chavela; Juan Manuel Sánchez-Cerritos

doi:10.3934/cpaa.2021090

Article Contents

2021, Volume 20, Issue 9: 2941-2963. Doi: 10.3934/cpaa.2021090

This issue Previous Article On the Calderon-Zygmund property of Riesz-transform type operators arising in nonlocal equations Next Article Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D

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
^* Corresponding author

Received: May 2020

Revised: April 2021

Early access: June 2021

Published: September 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37C83, 34C14; Secondary: 70F10, 70F15.

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Figure 1. Phase space for $ y(t),\dot{y}(t) $ ($ x(t) = 0 $)

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Figure 2. The vertical case

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Figure 3. Phase portrait in non-regularized and regularized coordinates

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Figure 4. The geodesic case

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Figure 5. The geodesic case, with $ M $ between $ \mu $ and $ m_1 $

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Figure 6. Phase portrait in n non-regularized and regularized coordinates

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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 $

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Figure 8. An orbit in the $ xy $-plane and its corresponding in the regularized plane

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Figure 9. An orbit in the $ xy $-plane and its corresponding in the regularized plane

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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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