The global phase space for the 2- and 3-dimensional Kepler problems
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.
scientific article; zbMATH DE number 5664403
scientific article