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Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature

Published:2016-06-17
Printed: Oct 2017
Departamento de Matemática, Facultad de Ciencias, Universidad de Bio-Bio, Casilla 5--C, Concepción, VIII--región, Chile
• Nestor Dávila,
Departamento de Matemática, Facultad de Ciencias, Universidad de Bio-Bio, Casilla 5--C, Concepción, VIII--región, Chile
• Ernesto Pérez-Chavela,
Departamento de MatemÃ¡ticas, Instituto Tecnológico Autónomo de México, (ITAM), Río Hondo 1, Col. Progreso Tizapán, Ciudad de México, 01080, México
• Claudio Vidal,
Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad de Bio-Bio, Casilla 5--C, Concepción, VIII--región, Chile
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Abstract

We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, $\mathbb S^2$ and $\mathbb H^2$, respectively) as function of the angular momentum and the energy. Hill's region are characterized and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum and we describe the orbits of the regularized vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$ are pointed out.
 Keywords: Kepler problem on surfaces of constant curvature, Hill's region, singularities, regularization, qualitative analysis of ODE
 MSC Classifications: 70F16 - Collisions in celestial mechanics, regularization 70G60 - Dynamical systems methods

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