19 research outputs found
Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere and the hyperbolic plane
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 , or on the hyperbolic plane , 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 .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
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