The global phase space for the 2- and 3-dimensional Kepler problems
type: Item
instance of: scholarly article
title: The global phase space for the 2- and 3-dimensional Kepler problems
zbMATH Open document ID: 1190.70005
author: Márcia P. Dantas
author: Jaume Llibre
published in: Qualitative Theory of Dynamical Systems
publication date: 2010-01-29T00:00:00Z
review text: This paper provides an extensive investigation of four specific integrable Hamiltonian systems derived from the Kepler problem. The focus of the study is on foliations of the phase space for each system. The four systems that are studied are the sidereal Kepler problem (in two and three dimensions) and the synodical Kepler problem (also in two and three dimensions). The sidereal Kepler problem is associated with the Hamiltonian \[ H= {1\over 2}\| y\|^2-{1\over\| x\|}, \] where \((x,y)\in (\mathbb{R}^3\setminus\{0\})\times\mathbb{R}^3\). This system has three independent integrals in involution: the energy, the third component of angular momentum, and the magnitude of angular momentum. The planar version of the problem is defined analogously. The three-dimensional synodical Kepler problem has Hamiltonian \[ H= {1\over 2}\| y\|^2- {1\over\| x\|}- x_1 y_2+ x_2 y_1 \] with \((x,y)\in (\mathbb{R}^3\setminus \{0\})\times\mathbb{R}^3\). This system also has three integrals in involution: the Jacobi integral, the sidereal energy, and the magnitude of the angular momentum. The planar synodical problem is defined similarly. The authors' exposition begins with background material in the theory of Hamiltonian system. Next they consider the sidereal Kepler problem. They study the critical values and the regions of motion, and then investigate the topology of invariant sets. After regularizing singularities, the authors establish foliations of the compactified phase space for both planar and spatial versions of the problem. The synodical Kepler problem is treated similarly.
Mathematics Subject Classification ID: 70F05
Mathematics Subject Classification ID: 70G10
Mathematics Subject Classification ID: 70H06
zbMATH DE Number: 5664403
zbMATH Keywords: sidereal Kepler problem
zbMATH Keywords: synodical Kepler problem
zbMATH Keywords: foliations
zbMATH Keywords: compactified phase space
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
full work available at URL: s12346-009-0002-0
OpenAlex ID: W2072455118
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DOI: 10.1007/S12346-009-0002-0
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description: scientific article; zbMATH DE number 5664403