|
SIGMA 15 (2019), 037, 30 pages arXiv:1810.13368
https://doi.org/10.3842/SIGMA.2019.037
Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems
Allan P. Fordy a and Qing Huang b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Northwest University, Xi'an 710069, People's Republic of China
Received November 01, 2018, in final form April 16, 2019; Published online May 05, 2019
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.
Key words: Darboux-Koenigs metrics; Hamiltonian system; super-integrability; Poisson algebra; conformal algebra.
pdf (500 kb)
tex (30 kb)
References
- Ballesteros A., Enciso A., Herranz F.J., Ragnisco O., Superintegrability on $N$-dimensional curved spaces: central potentials, centrifugal terms and monopoles, Ann. Physics 324 (2009), 1219-1233, arXiv:0812.1882.
- Ballesteros A., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability, Ann. Physics 326 (2011), 2053-2073, arXiv:1102.5494.
- Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry - methods and applications, Vols. 1-3, Springer-Verlag, New York, 1984.
- Escobar-Ruiz M.A., Miller Jr. W., Toward a classification of semidegenerate 3D superintegrable systems, J. Phys. A: Math. Theor. 50 (2017), 095203, 22 pages, arXiv:1611.02977.
- Fordy A.P., A Kaluza-Klein reduction of super-integrable systems, J. Geom. Phys. 131 (2018), 210-219, arXiv:1801.02981.
- Fordy A.P., First integrals from conformal symmetries: Darboux-Koenigs metrics and beyond, arXiv:1804.06904.
- Fordy A.P., Huang Q., Poisson algebras and 3D superintegrable Hamiltonian systems, SIGMA 14 (2018), 022, 37 pages, arXiv:1708.07024.
- Gilmore R., Lie groups, Lie algebras, and some of their applications, Wiley, New York, 1974.
- Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, arXiv:math-ph/0307039.
- Kalnins E.G., Kress J.M., Winternitz P., Superintegrability in a two-dimensional space of nonconstant curvature, J. Math. Phys. 43 (2002), 970-983, arXiv:math-ph/0108015.
- Koenigs G.X.P., Sur les géodésiques a integrales quadratiques, in Le cons sur la théorie générale des surfaces, Vol. 4, Editor J.G. Darboux, Chelsea Publishing, New York, 1972, 368-404.
- Kruglikov B., The D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math. 723 (2017), 153-215, arXiv:1303.1307.
- Matveev V.S., Shevchishin V.V., Two-dimensional superintegrable metrics with one linear and one cubic integral, J. Geom. Phys. 61 (2011), 1353-1377, arXiv:1010.4699.
- Miller Jr. W., Post S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A: Math. Theor. 46 (2013), 423001, 97 pages, arXiv:1309.2694.
- Valent G., Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree (I), Regul. Chaotic Dyn. 22 (2017), 319-352, arXiv:1703.10870.
|
|