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

SIGMA 14 (2018), 022, 37 pages      arXiv:1708.07024

Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Allan P. Fordy a and Qing Huang b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Northwest University, Xi'an 710069, People's Republic of China

Received August 24, 2017, in final form March 06, 2018; Published online March 16, 2018

Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras of the ''kinetic energy'', related to the quadratic Casimir function. We then consider the potentials which can be added, whilst remaining integrable, leading to families of separable systems, depending upon arbitrary functions of a single variable. Adding further integrals, in the superintegrable case, restricts these functions to specific forms, depending upon a finite number of arbitrary parameters. The Poisson algebras of these superintegrable systems are studied. The automorphisms of the symmetry algebra of the kinetic energy are extended to the full Poisson algebra, enabling us to build the full set of Poisson relations.

Key words: Hamiltonian system; super-integrability; Poisson algebra; conformal algebra; constant curvature.

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  1. Cariglia M., Hidden symmetries of dynamics in classical and quantum physics, Rev. Modern Phys. 86 (2014), 1283-1333, arXiv:1411.1262.
  2. Clarkson P.A., Olver P.J., Symmetry and the Chazy equation, J. Differential Equations 124 (1996), 225-246.
  3. Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055.
  4. Dubrovin B.A., Fomenko A.T., Novikov S.P., Modern geometry - methods and applications, Part I, The geometry of surfaces, transformation groups, and fields, Graduate Texts in Mathematics, Vol. 93, Springer-Verlag, New York, 1984.
  5. Eisenhart L.P., Riemannian geometry, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.
  6. Escobar-Ruiz M.A., Miller Jr. W., Toward a classification of semidegenerate 3D superintegrable systems, J. Phys. A: Math. Theor. 50 (2017), 095203, 22 pages, arXiv:1611.02977.
  7. Fordy A.P., Quantum super-integrable systems as exactly solvable models, SIGMA 3 (2007), 025, 10 pages, math-ph/0702048.
  8. Fordy A.P., A note on some superintegrable Hamiltonian systems, J. Geom. Phys. 115 (2017), 98-103, arXiv:1601.03079.
  9. Fordy A.P., Scott M.J., Recursive procedures for Krall-Sheffer operators, J. Math. Phys. 54 (2013), 043516, 23 pages, arXiv:1211.3075.
  10. Gilmore R., Lie groups, Lie algebras, and some of their applications, Wiley, New York, 1974.
  11. Kalnins E.G., Kress J.M., Miller Jr. W., Fine structure for 3D second-order superintegrable systems: three-parameter potentials, J. Phys. A: Math. Theor. 40 (2007), 5875-5892.
  12. Landau L.D., Lifshitz E.M., Mechanics, Course of Theoretical Physics, Vol. 1, Mechanics, Pergamon Press, Oxford, 1976.
  13. Marshall I., Wojciechowski S., When is a Hamiltonian system separable?, J. Math. Phys. 29 (1988), 1338-1346.
  14. 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.
  15. Smirnov R.G., The classical Bertrand-Darboux problem, J. Math. Sci. 151 (2008), 3230-3244, math-ph/0604038.
  16. Whittaker E.T., A treatise on the analytical dynamics of particles and rigid bodies, Cambridge University Press, Cambridge, 1988.

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