Fordy, AP orcid.org/0000-0002-2523-0262 and Huang, Q (2019) Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 15. 37. ISSN 1815-0659
Abstract
The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | This is an open access article under the terms of the Creative Commons Attribution-ShareAlike License (CC-BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0/) . |
Keywords: | Darboux-Koenigs metrics; Hamiltonian system; super-integrability; Poisson algebra; conformal algebra |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 25 Apr 2019 11:28 |
Last Modified: | 05 Jul 2019 01:31 |
Status: | Published |
Publisher: | National Academy of Science of Ukraine |
Identification Number: | 10.3842/SIGMA.2019.037 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:145314 |