Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2S^2 and the hyperbolic plane H2H^2

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature \k as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for \k>0, or \k<0, respectively; in addition, the Euclidean case arises as the particular case \k=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods: first by direct integration and second by obtaining the \k-dependent version of the Binet's equation. The final part of the article, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.Comment: 37 pages, 7 figure

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimen\-sional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.Comment: LaTeX 60 pages, DESY 94-01

Classical and quantum integrability in 3D systems

In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in $S^3$.Comment: plenary talk on the Conference QTS-5, July 2007, Valladolid, Spai

Path Integration and Separation of Variables in Spaces of Constant Curvature in Two and Three Dimensions

In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e.\ the flat spaces \bbbr^2 and \bbbr^3, the two- and three-dimensional sphere and the two- and three dimensional pseudosphere. The Laplace operator in these spaces admits separation of variables in various coordinate systems. In all these coordinate systems the path integral formulation will be stated, however in most of them an explicit solution in terms of the spectral expansion can be given only on a formal level. What can be stated in all cases, are the propagator and the corresponding Green function, respectively, depending on the invariant distance which is a coordinate independent quantity. This property gives rise to numerous identities connecting the corresponding path integral representations and propagators in various coordinate systems with each other.Comment: 70 pages, AmSTeX, DESY 93 - 141 (mailer corrupted file, and truncated it