Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Ángel Ballesteros ; Alberto Enciso ; Francisco J. Herranz ; Orlando Ragnisco ; Danilo Riglioni

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

The Stäckel transform is applied to the geodesic motion on Euclidean space, through the harmonic oscillator and Kepler–Coloumb potentials, in order to obtain maximally superintegrable classical systems on $N$-dimensional Riemannian spaces of nonconstant curvature. By one hand, the harmonic oscillator potential leads to two families of superintegrable systems which are interpreted as an intrinsic Kepler–Coloumb system on a hyperbolic curved space and as the so-called Darboux III oscillator. On the other, the Kepler–Coloumb potential gives rise to an oscillator system on a spherical curved space as well as to the Taub-NUT oscillator. Their integrals of motion are explicitly given. The role of the (flat/curved) Fradkin tensor and Laplace–Runge–Lenz $N$-vector for all of these Hamiltonians is highlighted throughout the paper. The corresponding quantum maximally superintegrable systems are also presented.

Détail

Keywords: coupling constant metamorphosis; integrable systems; curvature; harmonic oscillator; Kepler–Coulomb; Fradkin tensor; Laplace–Runge–Lenz vector; Taub-NUT; Darboux surfaces.

Bibliographie

[1] Hietarinta J., Grammaticos B., Dorizzi B., Ramani A., “Coupling-constant metamorphosis and duality between integrable Hamiltonian systems”, Phys. Rev. Lett., 53 (1984), 1707–1710 | DOI | MR

[2] Boyer C. P., Kalnins E. G., Miller W. Jr., “Stäckel-equivalent integrable Hamiltonian systems”, SIAM J. Math. Anal., 17 (1986), 778–797 | DOI | MR | Zbl

[3] Kalnins E. G., Kress J. M., Miller W. Jr., “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

[4] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stackel transform and 3D classification theory”, J. Math. Phys., 47 (2006), 043514, 26 pp. | DOI | MR | Zbl

[5] 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

[6] Daskaloyannis C., Tanoudis Y., “Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms”, Phys. Atom. Nuclei, 71 (2008), 853–861 | DOI

[7] Kalnins E. G., Miller W. Jr., 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

[8] Fradkin D. M., “Three-dimensional isotropic harmonic oscillator and $SU_3$”, Amer. J. Phys., 33 (1965), 207–211 | DOI | MR | Zbl

[9] Demkov Yu N., “Symmetry group of the isotropic oscillator”, Soviet Phys. JETP, 9 (1959), 63–66 | MR | Zbl

[10] Ballesteros A., Enciso A., Herranz F. J., Ragnisco O., “Hamiltonian systems admitting a Runge–Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds”, Comm. Math. Phys., 290 (2009), 1033–1049, arXiv: 0810.0999 | DOI | MR | Zbl

[11] Ballesteros A., 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

[12] Ngome J.-P., “Curved manifolds with conserved Runge–Lenz vectors”, J. Math. Phys., 50 (2009), 122901, 13 pp., arXiv: 0908.1204 | DOI | MR | Zbl

[13] Ballesteros A., Enciso A., Herranz F. J., Ragnisco O., “A maximally superintegrable system on an $n$-dimensional space of nonconstant curvature”, Phys. D, 237 (2008), 505–509, arXiv: math-ph/0612080 | DOI | MR | Zbl

[14] 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 | DOI | MR | Zbl

[15] Ballesteros A., Enciso A., Herranz F. J., Ragnisco O., Riglioni D., “On two superintegrable nonlinear oscillators in $N$ dimensions”, Int. J. Theor. Phys. (to appear) , arXiv: 1010.3358 | DOI

[16] Gibbons G. W., Manton N. S., “Classical and quantum dynamics of BPS monopoles”, Nuclear Phys. B, 274 (1986), 183–224 | DOI | MR

[17] Fehér L. G., Horváthy P. A., “Dynamic symmetry of monopole scattering”, Phys. Lett. B, 183 (1987), 182–186 | DOI | MR

[18] Gibbons G. W., Ruback P. J., “The hidden symmetries of multi-centre metrics”, Comm. Math. Phys., 115 (1988), 267–300 | DOI | MR | Zbl

[19] Iwai T., Katayama N., “Two kinds of generalized Taub-NUT metrics and the symmetry of associated dynamical systems”, J. Phys. A: Math. Gen., 27 (1994), 3179–3190 | DOI | MR | Zbl

[20] Iwai T., Katayama N., “Multifold Kepler systems – dynamical systems all of whose bounded trajectories are closed”, J. Math. Phys., 36 (1995), 1790–1811 | DOI | MR | Zbl

[21] Bini D., Cherubini C., Jantzen R. T., “Circular holonomy in the Taub-NUT spacetime”, Classical Quantum Gravity, 19 (2002), 5481–5488, arXiv: gr-qc/0210003 | DOI | MR | Zbl

[22] Jezierski J., Lukasik M., “Conformal Yano–Killing tensors for the Taub-NUT metric”, Classical Quantum Gravity, 24 (2007), 1331–1340, arXiv: gr-qc/0610090 | DOI | MR | Zbl

[23] Gibbons G. W., Warnick C. M., “Hidden symmetry of hyperbolic monopole motion”, J. Geom. Phys., 57 (2007), 2286–2315, arXiv: hep-th/0609051 | DOI | MR | Zbl

[24] Evans N. W., “Superintegrability in classical mechanics”, Phys. Rev. A, 41 (1990), 5666–5676 | DOI | MR

[25] Grosche C., Pogosyan G. S., Sissakian A. N., “Path integral approach for superintegrable potentials on the three-dimensional hyperboloid”, Phys. Part. Nuclei, 28 (1997), 486–519 | DOI

[26] Kalnins E. G., Miller W. Jr., Pogosyan G. S., “Superintegrability of the two-dimensional hyperboloid”, J. Math. Phys., 38 (1997), 5416–5433 | DOI | MR | Zbl

[27] Kalnins E. G., Miller W. Jr., Hakobyan Ye. M., Pogosyan G. S., “Superintegrability on the two-dimensional hyperboloid. II”, J. Math. Phys., 40 (1999), 2291–2306, arXiv: quant-ph/9907037 | DOI | MR | Zbl

[28] Rañada M. F., Santander M., “Superintegrable systems on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$”, J. Math. Phys., 40 (1999), 5026–5057 | DOI | MR | Zbl

[29] Kalnins E. G., Miller W. Jr., Pogosyan G. S., “Completeness of multiseparable superintegrability on the complex 2-sphere”, J. Phys. A: Math. Gen., 33 (2000), 6791–6806 | DOI | MR | Zbl

[30] Kalnins E. G., Kress J. M., Pogosyan G. S., Miller W. Jr., “Completeness of superintegrability in two-dimensional constant-curvature spaces”, J. Phys. A: Math. Gen., 34 (2001), 4705–4720, arXiv: math-ph/0102006 | DOI | MR | Zbl

[31] Cariñena J. F., Rañada M. F., Santander M., Sanz-Gil T., “Separable potentials and a triality in two-dimensional spaces of constant curvature”, J. Nonlinear Math. Phys., 12 (2005), 230–252 | DOI | MR | Zbl

[32] Ballesteros A., Herranz F. J., “Universal integrals for superintegrable systems on $N$-dimensional spaces of constant curvature”, J. Phys. A: Math. Theor., 40 (2007), F51–F59, arXiv: math-ph/0610040 | DOI | MR | Zbl

[33] Ragnisco O., Ballesteros A., Herranz F. J., Musso F., “Quantum deformations and superintegrable motions on spaces with variable curvature”, SIGMA, 3 (2007), 026, 20 pp., arXiv: math-ph/0611040 | DOI | MR | Zbl

[34] Bertrand J., “Théorème relatif au mouvement d'un point attiré vers un centre fixe”, C.R. Math. Acad. Sci. Paris, 77 (1873), 849–853

[35] Ballesteros A., Enciso A., Herranz F. J., Ragnisco O., “$N$-dimensional $sl(2)$-coalgebra spaces with non-constant curvature”, Phys. Lett. B, 652 (2007), 376–383, arXiv: 0704.1470 | DOI | MR

[36] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 053509, 28 pp. | DOI | MR | Zbl

[37] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory”, J. Math. Phys., 46 (2005), 103507, 28 pp. | DOI | MR | Zbl

[38] Kalnins E. G., Kress J. M., Miller W. Jr., “Second order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems”, J. Math. Phys., 46 (2006), 093501, 25 pp. | DOI | MR

[39] Kustaanheimo P., Stiefel E., “Perturbation theory of Kepler motion based on spinor regularization”, J. Reine Angew. Math., 218 (1965), 204–219 | DOI | MR | Zbl

[40] Kozlov R., “A conservative discretization of the Kepler problem based on the $L$-transformations”, Phys. Lett. A, 369 (2007), 262–273 | DOI | MR | Zbl

[41] Koenigs G., “Sur les géodésiques a intégrales quadratiques”, Le{ç}ons sur la théorie générale des surfaces, 4, eds. G. Darboux, Chelsea, New York, 1972, 368–404

[42] Kalnins E. G., Kress J. M., Miller W. Jr., Winternitz P., “Superintegrable systems in Darboux spaces”, J. Math. Phys., 44 (2003), 5811–5848, arXiv: math-ph/0307039 | DOI | MR | Zbl

[43] Grosche C., Pogosyan G. S., Sissakian A. N., “Path integral approach for superintegrable potentials on spaces of nonconstant curvature. I. Darboux spaces $D_{\rm I}$ and $D_{\rm II}$”, Phys. Part. Nuclei, 38 (2007), 299–325, arXiv: quant-ph/0608083 | DOI

[44] Grosche C., Pogosyan G. S., Sissakian A. N., “Path integral approach for superintegrable potentials on spaces of nonconstant curvature. II. Darboux spaces $D_{\rm III}$ and $D_{\rm IV}$”, Phys. Part. Nuclei, 38 (2007), 525–563, arXiv: quant-ph/0609058 | DOI

[45] Ballesteros A., Enciso A., Herranz F. J., Ragnisco O., Riglioni D., “A new exactly solvable quantum model in $N$ dimensions”, Phys. Lett. A, 375 (2011), 1431–1435, arXiv: 1007.1335 | DOI | MR

[46] Iwai T., Uwano Y., Katayama N., “Quantization of the multifold Kepler system”, J. Math. Phys., 37 (1996), 608–624 | DOI | MR | Zbl

[47] 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 (to appear) , arXiv: 1102.5494 | DOI | MR

[48] Ragnisco O., Riglioni D., “A family of exactly solvable radial quantum systems on space of non-constant curvature with accidental degeneracy in the spectrum”, SIGMA, 6 (2010), 097, 10 pp., arXiv: 1010.0641 | DOI | MR | Zbl