Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 010, 22 pages      math-ph/0512084      https://doi.org/10.3842/SIGMA.2006.010

Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature

Francisco José Herranz a and Ángel Ballesteros b
a) Departamento de Física, Escuela Politécnica Superior, Universidad de Burgos, 09001 Burgos, Spain
b) Departamento de Física, Facultad de Ciencias, Universidad de Burgos, 09001 Burgos, Spain

Received December 21, 2005, in final form January 20, 2006; Published online January 24, 2006

Abstract
A family of classical superintegrable Hamiltonians, depending on an arbitrary radial function, which are defined on the 3D spherical, Euclidean and hyperbolic spaces as well as on the (2+1)D anti-de Sitter, Minkowskian and de Sitter spacetimes is constructed. Such systems admit three integrals of the motion (besides the Hamiltonian) which are explicitly given in terms of ambient and geodesic polar coordinates. The resulting expressions cover the six spaces in a unified way as these are parametrized by two contraction parameters that govern the curvature and the signature of the metric on each space. Next two maximally superintegrable Hamiltonians are identified within the initial superintegrable family by finding the remaining constant of the motion. The former potential is the superposition of a (curved) central harmonic oscillator with other three oscillators or centrifugal barriers (depending on each specific space), so that this generalizes the Smorodinsky-Winternitz system. The latter one is a superposition of the Kepler-Coulomb potential with another two oscillators or centrifugal barriers. As a byproduct, the Laplace-Runge-Lenz vector for these spaces is deduced. Furthermore both potentials are analysed in detail for each particular space. Some comments on their generalization to arbitrary dimension are also presented.

Key words: integrable systems; curvature; contraction; harmonic oscillator; Kepler-Coulomb; hyperbolic; de Sitter.

pdf (336 kb)   ps (238 kb)   tex (25 kb)

References

  1. Ballesteros A., Herranz F.J., Integrable deformations of oscillator chains from quantum algebras, J. Phys. A: Math. Gen., 1999, V.32, N 50, 8851-8862, solv-int/9911004.
  2. Ballesteros A., Herranz F.J., Musso F., Ragnisco O., Superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian, in Superintegrability in Classical and Quantum Systems, Editors P. Tempesta, P. Winternitz, J. Harnad, W. Miller Jr., G. Pogosyan and M.A. Rodriguez, CRM Proceedings and Lecture Notes, Providence, American Mathematical Society, 2004, V.37, 1-14, math-ph/0412067.
  3. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., Quantum structure of the motion groups of the two-dimensional Cayley-Klein geometries, J. Phys. A: Math. Gen., 1993, V.26, N 21, 5801-5823.
  4. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., Quantum (2+1) kinematical algebras: a global approach, J. Phys. A: Math. Gen., 1994, V.27, N 4, 1283-1297.
  5. Ballesteros A., Herranz F.J., del Olmo M.A., Santander M., Classical deformations, Poisson-Lie contractions, and quantization of dual Lie bialgebras, J. Math. Phys., 1995, V.36, N 2, 631-640.
  6. Ballesteros A., Herranz F.J., Ragnisco O., Curvature from quantum deformations, Phys. Lett. B, 2005, V.610, N 1-2, 107-114, hep-th/0504065.
  7. Ballesteros A., Herranz F.J., Ragnisco O., Integrable geodesic motion on 3D curved spaces from non-standard quantum deformations, Czech. J. Phys., 2005, V.55, N 11, 1327-1333, math-ph/0508038.
  8. Ballesteros A., Herranz F.J., Ragnisco O., Integrable potentials on spaces with curvature from quantum groups, J. Phys. A: Math. Gen., 2005, V.38, N 32, 7129-7144, math-ph/0505081.
  9. Ballesteros A., Herranz F.J., Santander M., Sanz-Gil T., Maximal superintegrability on N-dimensional curved spaces, J. Phys. A: Math. Gen., 2003, V.36, N 7, L93-L99, math-ph/0211012.
  10. Ballesteros A., Ragnisco O., A systematic construction of integrable Hamiltonians from coalgebras, J. Phys. A: Math. Gen., 1998, V.31, N 16, 3791-3813, solv-int/9802008.
  11. Cariñena J.F., Rañada M.F., Santander M., Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys., 2005, V.46, N 5, 052702, 18 pages, math-ph/0504016.
  12. Cariñena J.F., Rañada M.F., Santander M., Sanz-Gil T., Separable potentials and a triality in two-dimensionl spaces of constant curvature, J. Nonlinear Math. Phys., 2005, V.12, N 2, 230-252.
  13. Doubrovine B., Novikov S., Fomenko A., Géométrie Contemporaine, Méthodes et Applications, Part 1, Traduit du Russe, Mathematiques, Moscow, Mir, 1985 (in French).
  14. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A, 1990, V.41, N 10, 5666-5676.
  15. Evans N.W., Superintegrability of the Winternitz system, Phys. Lett. A, 1990, V.147, N 8-9, 483-486.
  16. Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys., 1991, V.32, N 12, 3369-3375.
  17. Fris J., Mandrosov V., Smorodinsky Y.A., Uhlir M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett., 1965, V.16, N 3, 354-356.
  18. Gromov N.A., Man'ko V.I., The Jordan-Schwinger representations of Cayley-Klein groups. I. The orthogonal groups, J. Math. Phys., 1990, V.31, N 5, 1047-1053.
  19. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials 1. Two- and three-dimensional Euclidean space, Fortschr. Phys., 1995, V.43, N 6, 453-521, hep-th/9402121.
  20. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral discussion for Smorodinsky-Winternitz potentials 2. The two- and three-dimensional sphere, Fortschr. Phys., 1995, V.43, N 6, 523-563.
  21. Grosche C., Pogosyan G.S., Sissakian A.N., Path integral approach for superintegrable potentials on the three-dimensional hyperboloid, Phys. Part. Nuclei, 1997, V.28, N 5, 486-519.
  22. Helgason S., Differential geometry and symmetric spaces, New York, Academic Press, 1962.
  23. Herranz F.J., Ballesteros A., Santander M., Sanz-Gil T., Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces, in Superintegrability in Classical and Quantum Systems, Editors P. Tempesta, P. Winternitz, J. Harnad, W. Miller Jr., G. Pogosyan and M.A. Rodriguez, CRM Proceedings and Lecture Notes, Providence, American Mathematical Society, 2004, V.37, 75-89, math-ph/0501035.
  24. Herranz F.J., Ortega R., Santander M., Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A: Math. Gen., 2000, V.33, N 24, 4525-4551, for an extended version see math-ph/9910041.
  25. Herranz F.J., Santander M., Conformal symmetries of spacetimes, J. Phys. A: Math. Gen., 2002, V.35, N 31, 6601-6618; math-ph/0110019.
  26. Higgs P.W., Dynamical symmetries in a spherical geometry I, J. Phys. A: Math. Gen., 1979, V.12, N 3, 309-323.
  27. Izmest'ev A.A., Pogosyan G.S., Sissakian A.N., Winternitz P., Contractions of Lie algebras and separation of variables. The n-dimensional sphere, J. Math. Phys., 1999, V.40, N 3, 1549-1573.
  28. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen., 2001, V.34, N 22, 4705-4720, math-ph/0102006.
  29. Kalnins E.G., Miller W., Hakobyan Y.M., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid II, J. Math. Phys., 1999, V.40, N 5, 2291-2306, quant-ph/9907037.
  30. Kalnins E.G., Miller W., Pogosyan G.S., Superintegrability of the two-dimensional hyperboloid, J. Math. Phys., 1997, V.38, N 10, 5416-5433.
  31. Kalnins E.G., Miller W., Pogosyan G.S., Completeness of multiseparable superintegrability on the complex 2-sphere, J. Phys. A: Math. Gen., 2000, V.33, N 38, 6791-6806.
  32. Kalnins E.G., Miller W., Pogosyan G.S., Coulomb-oscillator duality in spaces of constant curvature, J. Math. Phys., 2000, V.41, N 5, 2629-2657, quant-ph/9906055.
  33. Kalnins E.G., Miller W., Pogosyan G.S., The Coulomb-oscillator relation on n-dimensional spheres and hyperboloids, Phys. Atomic Nuclei, 2002, V.65, N 6, 1086-1094, math-ph/0210002.
  34. Kalnins E.G., Pogosyan G.S., Miller W., Completeness of multiseparable superintegrability in two dimensions, Phys. Atomic Nuclei, 2002, V.65, N 6, 1033-1035.
  35. Kalnins E.G., Williams G.C., Miller W., Pogosyan G.S., Superintegrability in three-dimensional Euclidean space, J. Math. Phys., 1999, V.40, N 2, 708-725.
  36. Leemon H.I., Dynamical symmetries in a spherical geometry II, J. Phys. A: Math. Gen., 1979, V.12, N 4, 489-501.
  37. Nersessian A., Pogosyan G., Relation of the oscillator and Coulomb systems on spheres and pseudospheres, Phys. Rev. A, 2001, V.63, N 2, 020103, quant-ph/0006118.
  38. Perelomov A.M., Integrable systems of classical mechanics and Lie algebras, Berlin, Birkhäuser, 1990.
  39. Rañada M.F., Santander M., Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys., 1999, V.40, N 10, 5026-5057.
  40. Rañada M.F., Santander M., On some properties of harmonic oscillator on spaces of constant curvature, Rep. Math. Phys., 2002, V.49, N 2-3, 335-343.
  41. Rañada M.F., Santander M., On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys., 2002, V.43, N 1, 431-451.
  42. Rañada M. F., Santander M., On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2 II, J. Math. Phys., 2003, V.44, N 5, 2149-2167.
  43. Schrödinger E., A method of determining quantum mechanical eigenvalues and eigenfunctions, Proc. R. Ir. Acad. A, 1940, V.46, 9-16.


Previous article   Next article   Contents of Volume 2 (2006)