Structure theory for extended Kepler–Coulomb 3D classical superintegrable systems

Ernie G. Kalnins ; Willard Miller Jr.

Symmetry, integrability and geometry: methods and applications, Tome 8 (2012)

Résumé

The classical Kepler–Coulomb system in 3 dimensions is well known to be 2nd order superintegrable, with a symmetry algebra that closes polynomially under Poisson brackets. This polynomial closure is typical for 2nd order superintegrable systems in 2D and for 2nd order systems in 3D with nondegenerate (4-parameter) potentials. However the degenerate 3-parameter potential for the 3D extended Kepler–Coulomb system (also 2nd order superintegrable) is an exception, as its quadratic symmetry algebra doesn't close polynomially. The 3D 4-parameter potential for the extended Kepler–Coulomb system is not even 2nd order superintegrable. However, Verrier and Evans (2008) showed it was 4th order superintegrable, and Tanoudis and Daskaloyannis (2011) showed that in the quantum case, if a second 4th order symmetry is added to the generators, the double commutators in the symmetry algebra close polynomially. Here, based on the Tremblay, Turbiner and Winternitz construction, we consider an infinite class of classical extended Kepler–Coulomb 3- and 4-parameter systems indexed by a pair of rational numbers $(k_1,k_2)$ and reducing to the usual systems when $k_1=k_2=1$. We show these systems to be superintegrable of arbitrarily high order and work out explicitly the structure of the symmetry algebras determined by the 5 basis generators we have constructed. We demonstrate that the symmetry algebras close rationally; only for systems admitting extra discrete symmetries is polynomial closure achieved. Underlying the structure theory is the existence of raising and lowering constants of the motion, not themselves polynomials in the momenta, that can be employed to construct the polynomial symmetries and their structure relations.

Détail

Keywords: superintegrability
Mots-clés : Kepler–Coulomb system.

Bibliographie

[1] Ballesteros Á., Herranz F.J., “Maximal superintegrability of the generalized Kepler–Coulomb system on $N$-dimensional curved spaces”, J. Phys. A: Math. Theor., 42 (2009), 245203, 12 pp. ; arXiv: 0903.2337 | DOI | MR | Zbl

[2] Eastwood M., Miller W. (Editors), Symmetries and overdetermined systems of partial differential equations, The IMA Volumes in Mathematics and its Applications, 144, Springer, New York, 2008 | DOI | MR

[3] Evans N.W., Verrier P.E., “Superintegrability of the caged anisotropic oscillator”, J. Math. Phys., 49 (2008), 092902, 10 pp. ; arXiv: 0808.2146 | DOI | MR | Zbl

[4] Gonera C., Note on superintegrability of TTW model, arXiv: 1010.2915

[5] Kalnins E.G., Kress J.M., Miller W., “A recurrence relation approach to higher order quantum superintegrability”, SIGMA, 7 (2011), 031, 24 pp. ; arXiv: 1011.6548 | DOI | MR | Zbl

[6] Kalnins E.G., Kress J.M., Miller W., “Families of classical subgroup separable superintegrable systems”, J. Phys. A: Math. Theor., 43 (2010), 092001, 8 pp. ; arXiv: 0912.3158 | DOI | MR | Zbl

[7] Kalnins E.G., Kress J.M., Miller W., “Fine structure for 3D second-order superintegrable systems: three-parameter potentials”, J. Phys. A: Math. Theor., 40 (2007), 5875–5892 | DOI | MR | Zbl

[8] Kalnins E.G., Kress J.M., Miller W., “Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform”, J. Math. Phys., 46 (2005), 053510, 15 pp. | DOI | MR | Zbl

[9] Kalnins E.G., Kress J.M., Miller W., “Superintegrability and higher order integrals for quantum systems”, J. Phys. A: Math. Theor., 43 (2010), 265205, 21 pp. ; arXiv: 1002.2665 | DOI | MR | Zbl

[10] Kalnins E.G., Kress J.M., Miller W., “Tools for verifying classical and quantum superintegrability”, SIGMA, 6 (2010), 066, 23 pp. ; arXiv: 1006.0864 | DOI | MR | Zbl

[11] Kalnins E.G., Kress J.M., Miller W., Post S., “Structure theory for second order 2D superintegrable systems with 1-parameter potentials”, SIGMA, 5 (2009), 008, 24 pp. ; arXiv: 0901.3081 | DOI | MR | Zbl

[12] Kalnins E.G., Miller W., Pogosyan G.S., “Superintegrability and higher order constants for classical and quantum systems”, Phys. Atomic Nuclei, 74 (2011), 914–918 ; arXiv: 0912.2278 | DOI

[13] Kalnins E.G., Miller W., Post S., “Coupling constant metamorphosis and $N$th-order symmetries in classical and quantum mechanics”, J. Phys. A: Math. Theor., 43 (2010), 035202, 20 pp. ; arXiv: 0908.4393 | DOI | MR | Zbl

[14] Kalnins E.G., Miller W., Post S., “Models for quadratic algebras associated with second order superintegrable systems in 2D”, SIGMA, 4 (2008), 008, 21 pp. ; arXiv: 0801.2848 | DOI | MR | Zbl

[15] Kalnins E.G., Miller W., Post S., “Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere”, SIGMA, 7 (2011), 051, 26 pp. ; arXiv: 1010.3032 | DOI | MR | Zbl

[16] Maciejewski A.J., Przybylska M., Yoshida H., “Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces”, J. Phys. A: Math. Theor., 43 (2010), 382001, 15 pp. ; arXiv: 1004.3854 | DOI | MR | Zbl

[17] Maciejewski A.J., Przybylska M., Yoshida H., “Necessary conditions for the existence of additional first integrals for Hamiltonian systems with homogeneous potential”, Nonlinearity, 25 (2012), 255–277 ; arXiv: nlin.SI/0701057 | DOI | MR | Zbl

[18] Marquette I., “Construction of classical superintegrable systems with higher order integrals of motion from ladder operators”, J. Math. Phys., 51 (2010), 072903, 9 pp. ; arXiv: 1002.3118 | DOI | MR

[19] Post S., Winternitz P., “An infinite family of superintegrable deformations of the Coulomb potential”, J. Phys. A: Math. Theor., 43 (2010), 222001, 11 pp. ; arXiv: 1003.5230 | DOI | MR | Zbl

[20] Sergyeyev A., Blaszak M., “Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems”, J. Phys. A: Math. Theor., 41 (2008), 105205, 20 pp. ; arXiv: 0706.1473 | DOI | MR | Zbl

[21] Tanoudis Y., Daskaloyannis C., “Algebraic calculation of the energy eigenvalues for the nondegenerate three-dimensional Kepler–Coulomb potential”, SIGMA, 7 (2011), 054, 11 pp. ; arXiv: 1102.0397 | DOI | MR | Zbl

[22] Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (Editors), Superintegrability in classical and quantum systems, CRM Proceedings and Lecture Notes, 37, American Mathematical Society, Providence, RI, 2004 | MR

[23] Tremblay F., Turbiner A.V., Winternitz P., “An infinite family of solvable and integrable quantum systems on a plane”, J. Phys. A: Math. Theor., 42 (2009), 242001, 10 pp. ; arXiv: 0904.0738 | DOI | MR | Zbl

[24] Tremblay F., Turbiner A.V., Winternitz P., “Periodic orbits for an infinite family of classical superintegrable systems”, J. Phys. A: Math. Theor., 43 (2010), 015202, 14 pp. ; arXiv: 0910.0299 | DOI | MR | Zbl

[25] Tsiganov A.V., “Addition theorems and the Drach superintegrable systems”, J. Phys. A: Math. Theor., 41 (2008), 335204, 16 pp. ; arXiv: 0805.3443 | DOI | MR | Zbl

[26] Tsiganov A.V., “Leonard Euler: addition theorems and superintegrable systems”, Regul. Chaotic Dyn., 14 (2009), 389–406 ; arXiv: 0810.1100 | DOI | MR | Zbl

[27] Tsiganov A.V., “On maximally superintegrable systems”, Regul. Chaotic Dyn., 13 (2008), 178–190 ; arXiv: 0711.2225 | DOI | MR | Zbl

[28] Verrier P.E., Evans N.W., “A new superintegrable Hamiltonian”, J. Math. Phys., 49 (2008), 022902, 8 pp. ; arXiv: 0712.3677 | DOI | MR | Zbl