Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

DOI

10.2991/jnmp.2005.12.2.6How to use a DOI?

Abstract

We characterize and completely describe some types of separable potentials in twdimensional spaces, S2 [1]2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S2 , hyperbolic plane H2 , AntiDeSiter / DeSitter two-dimensional spaces AdS1+1 / dS1+1 as well as for their flat analogues E2 and M1+1 . This is achieved through an approach of Cayley-Klein type with two parameters, 1 and 2, to encompass all curvatures and signature types. We dicuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in S2 [1]2 and relate them by a formal triality transformation, which seems to be a clue to introduce general "elliptic coordinates" for any CK space concisely. As an application we give, in any S2 [1]2 , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any S2 [1]2 .

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - José F. Cariñena
AU  - Manuel F. Rañada
AU  - Mariano Santander
PY  - 2005
DA  - 2005/05/01
TI  - Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature
JO  - Journal of Nonlinear Mathematical Physics
SP  - 230
EP  - 252
VL  - 12
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2005.12.2.6
DO  - 10.2991/jnmp.2005.12.2.6
ID  - Cariñena2005
ER  -