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


SIGMA 11 (2015), 088, 51 pages      arXiv:1505.00527      http://dx.doi.org/10.3842/SIGMA.2015.088
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Examples of Complete Solvability of 2D Classical Superintegrable Systems

Yuxuan Chen a, Ernie G. Kalnins b, Qiushi Li a and Willard Miller Jr. a
a) School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, USA
b) Department of Mathematics, University of Waikato, Hamilton, New Zealand

Received May 05, 2015, in final form October 27, 2015; Published online November 03, 2015

Abstract
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved algebraically. In this paper we show explicitly, mostly through examples of 2nd order superintegrable systems in 2 dimensions, how the trajectories can be determined in detail using rather elementary algebraic, geometric and analytic methods applied to the closed quadratic algebra of symmetries of the system, without resorting to separation of variables techniques or trying to integrate Hamilton's equations. We treat a family of 2nd order degenerate systems: oscillator analogies on Darboux, nonzero constant curvature, and flat spaces, related to one another via contractions, and obeying Kepler's laws. Then we treat two 2nd order nondegenerate systems, an analogy of a caged Coulomb problem on the 2-sphere and its contraction to a Euclidean space caged Coulomb problem. In all cases the symmetry algebra structure provides detailed information about the trajectories, some of which are rather complicated. An interesting example is the occurrence of ''metronome orbits'', trajectories confined to an arc rather than a loop, which are indicated clearly from the structure equations but might be overlooked using more traditional methods. We also treat the Post-Winternitz system, an example of a classical 4th order superintegrable system that cannot be solved using separation of variables. Finally we treat a superintegrable system, related to the addition theorem for elliptic functions, whose constants of the motion are only rational in the momenta. It is a system of special interest because its constants of the motion generate a closed polynomial algebra. This paper contains many new results but we have tried to present most of the materials in a fashion that is easily accessible to nonexperts, in order to provide entrée to superintegrablity theory.

Key words: superintegrable systems; classical trajectories.

pdf (5574 kb)   tex (5224 kb)

References

  1. Arnold V.I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York - Heidelberg, 1978.
  2. Ballesteros Á., 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.
  3. Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Superintegrable oscillator and Kepler systems on spaces of nonconstant curvature via the Stäckel transform, SIGMA 7 (2011), 048, 15 pages, arXiv:1103.4554.
  4. Ballesteros Á., Herranz F.J., Musso F., The anisotropic oscillator on the 2D sphere and the hyperbolic plane, Nonlinearity 26 (2013), 971-990, arXiv:1207.0071.
  5. Capel J.J., Kress J.M., Invariant classification of second-order conformally flat superintegrable systems, J. Phys. A: Math. Theor. 47 (2014), 495202, 33 pages, arXiv:1406.3136.
  6. Chanu C., Degiovanni L., Rastelli G., Superintegrable three-body systems on the line, J. Math. Phys. 49 (2008), 112901, 10 pages, arXiv:0802.1353.
  7. Curtis H.D., Orbital mechanics for engineering students, 3rd ed., Elsevier, 2013.
  8. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  9. Evans N.W., Superintegrability in classical mechanics, Phys. Rev. A 41 (1990), 5666-5676.
  10. Evans N.W., Verrier P.E., Superintegrability of the caged anisotropic oscillator, J. Math. Phys. 49 (2008), 092902, 10 pages, arXiv:0808.2146.
  11. Fordy A.P., Quantum super-integrable systems as exactly solvable models, SIGMA 3 (2007), 025, 10 pages, math-ph/0702048.
  12. Goldstein H., Poole C.P., Safko J.L., Classical mechanics, Addison-Wesley Press, Inc., Boston, 2001.
  13. Granovskii Y.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved space. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
  14. Higgs P.W., Dynamical symmetries in a spherical geometry. I, J. Phys. A: Math. Gen. 12 (1979), 309-323.
  15. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510-524.
  16. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  17. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  18. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory, J. Math. Phys. 46 (2005), 103507, 28 pages.
  19. Kalnins E.G., Kress J.M., Miller Jr. W., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
  20. Kalnins E.G., Kress J.M., Miller Jr. W., Families of classical subgroup separable superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 092001, 8 pages, arXiv:0912.3158.
  21. Kalnins E.G., Kress J.M., Miller Jr. W., Tools for verifying classical and quantum superintegrability, SIGMA 6 (2010), 066, 23 pages, arXiv:1006.0864.
  22. Kalnins E.G., Kress J.M., Miller Jr. W., Winternitz P., Superintegrable systems in Darboux spaces, J. Math. Phys. 44 (2003), 5811-5848, math-ph/0307039.
  23. Kalnins E.G., Kress J.M., Pogosyan G.S., Miller Jr. W., Completeness of superintegrability in two-dimensional constant-curvature spaces, J. Phys. A: Math. Gen. 34 (2001), 4705-4720, math-ph/0102006.
  24. Kalnins E.G., Miller Jr. W., Post S., Contractions of 2D 2nd order quantum superintegrable systems and the Askey scheme for hypergeometric orthogonal polynomials, SIGMA 9 (2013), 057, 28 pages, arXiv:1212.4766.
  25. Koenigs G., Sur les géodésiques à intégrales quadratiques, in Darboux G., Lecons sur la théorie générale des surfaces et les applications geométriques du calcul infinitesimal, Vol. 4, Chelsea, New York, 1972, 368-404.
  26. Kuru S., Negro J., 'Spectrum generating algebras' of classical systems: the Kepler-Coulomb potential, J. Phys. Conf. Ser. 343 (2012), 012063, 5 pages.
  27. Latini D., Ragnisco O., The classical Taub-Nut system: factorization, spectrum generating algebra and solution to the equations of motion, J. Phys. A: Math. Theor. 48 (2015), 175201, 13 pages, arXiv:1411.3571.
  28. Létourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Physics 243 (1995), 144-168.
  29. 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 pages, arXiv:1004.3854.
  30. Makarov A.A., Smorodinsky J.A., Valiev K., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries. Part I: The integrals of motion, Nuovo Cimento 52 (1967), 1061-1084.
  31. Marquette I., Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems, J. Math. Phys. 53 (2012), 012901, 12 pages, arXiv:1109.4471.
  32. Mayrand M., Vinet L., Hidden symmetries of two-dimensional systems with local degeneracies, J. Math. Phys. 33 (1992), 203-212.
  33. 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.
  34. Nehorošev N.N., Action-angle variables, and their generalizations, Trans. Moscow Math. Soc. 26 (1972), 180-198.
  35. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010.
  36. Post S., Winternitz P., A nonseparable quantum superintegrable system in 2D real Euclidean space, J. Phys. A: Math. Theor. 44 (2011), 162001, 8 pages, arXiv:1101.5405.
  37. Rodríguez M.A., Tempesta P., Winternitz P., Symmetry reduction and superintegrable Hamiltonian systems, J. Phys. Conf. Ser. 175 (2009), 012013, 8 pages.
  38. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257, hep-th/0011209.
  39. Tempesta P., Winternitz P., Harnad J., Miller W., Pogosyan G., Rodriguez M. (Editors), Superintegrability in classical and quantum systems, CRM Proceedings and Lecture Notes, Vol. 37, Amer. Math. Soc., Providence, RI, 2004.
  40. 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 pages, arXiv:0904.0738.
  41. 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 pages, arXiv:0910.0299.
  42. Tsiganov A.V., On maximally superintegrable systems, Regul. Chaotic Dyn. 13 (2008), 178-190, arXiv:0711.2225.

Previous article  Next article   Contents of Volume 11 (2015)