Superintegrable systems on 3 dimensional conformally flat spaces. (English) Zbl 1466.37048

Summary: We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system.


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70G65 Symmetries, Lie group and Lie algebra methods for problems in mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
17B63 Poisson algebras
Full Text: DOI arXiv


[1] Capel, J. J.; Kress, J. M., Invariant classification of second-order conformally flat superintegrable systems, J. Phys. A, 47, 495202, 33 (2014) · Zbl 1314.14064
[2] Capel, J. J.; Kress, J. M.; Post, S., Invariant classification and limits of maximally superintegrable systems in 3D, SIGMA Symmetry Integrability Geom. Methods Appl., 11, 038, 17 (2015) · Zbl 1318.33014
[3] Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P., Modern Geometry-Methods and Applications (3 Volumes) (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0529.53002
[4] Escobar-Ruiz, M. A.; Miller Jr, W., Toward a classification of semidegenerate 3D superintegrable systems, J. Phys. A, 50, 095203, 22 (2017) · Zbl 1361.81065
[5] Fish, C. D.; Jordan, D. A., Connected quantized weyl algebras and quantum cluster algebras, J. Pure Appl. Algebra, 222, 2374-2412 (2018) · Zbl 1417.16030
[6] Fordy, A. P., Mutation-periodic quivers, integrable maps and associated poisson algebras, Phil. Trans. R. Soc. A, 369, 1264-1279 (2011) · Zbl 1219.17020
[7] Fordy, A. P., A kaluza-klein reduction of super-integrable systems, J. Geom. Phys., 131, 210-219 (2018) · Zbl 1396.70018
[8] Fordy, A. P., First integrals from conformal symmetries: darboux-koenigs metrics and beyond, J. Geom. Phys., 145, 103475, 13 (2019) · Zbl 1430.37063
[9] Fordy, A. P.; Galajinsky, A., Eisenhart lift of 2-dimensional mechanics, Eur. Phys. J. C, 79, 301, 11 (2019)
[10] Fordy, A. P.; Huang, Q., Poisson algebras and 3d superintegrable hamiltonian systems, SIGMA Symmetry Integrability Geom. Methods Appl., 14, 022, 37 (2018) · Zbl 1416.17011
[11] Fordy, A. P.; Huang, Q., Generalised darboux-koenigs metrics and 3 dimensional superintegrable systems, SIGMA Symmetry Integrability Geom. Methods Appl., 15, 037, 30 (2019) · Zbl 1420.37038
[12] Kalnins, E. G.; Kress, J. M.; Miller Jr., W.; Winternitz, P., Superintegrable systems in darboux spaces, J. Math. Phys., 44, 5811-5848 (2003) · Zbl 1063.37050
[13] Kalnins, E. G.; Kress, J. M.; Winternitz, P., Superintegrability in a two-dimensional space of nonconstant curvature, J. Math. Phys., 43, 970-983 (2002) · Zbl 1059.37040
[14] G. Koenigs, Sur les géodésiques a intégrales quadratiques, in: G. Darboux (Ed.), in: Leçons sur la théorie générale des surfaces, Vol. 4, Chelsea, New York, 1972, pp. 368-404.
[15] Kruglikov, B.; The, D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math., 2017, 723, 153-215 (2014) · Zbl 1359.58019
[16] Matveev, V. S.; Shevchishin, V. V., Two-dimensional superintegrable metrics with one linear and one cubic integral, J. Geom. Phys., 61, 1353-1377 (2011) · Zbl 1218.53087
[17] Miller, W.; Post, S.; Winternitz, P., Classical and quantum superintegrability with applications, J. Phys. A, 46, 423001, 97 (2013) · Zbl 1276.81070
[18] Valent, G., Superintegrable models on riemannian surfaces of revolution with integrals of any integer degree (i), Regul. Chaotic Dyn., 22, 319-352 (2017) · Zbl 1387.30067
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