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

SIGMA 1 (2005), 004, 7 pages      nlin.SI/0511007

A System of n = 3 Coupled Oscillators with Magnetic Terms: Symmetries and Integrals of Motion

Manuel F. Rañada
Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received July 06, 2005, in final form September 16, 2005; Published online September 20, 2005

The properties of a system of n = 3 coupled oscillators with linear terms in the velocities (magnetic terms) depending in two parameters are studied. We proved the existence of a bi-Hamiltonian structure arising from a non-symplectic symmetry, as well the existence of master symmetries and additional integrals of motion (weak superintegrability) for certain particular values of the parameters.

Key words: non-symplectic symmetries; bi-Hamiltonian structures; master symmetries; cubic integrals.

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  1. Cariñena J.F., Ibort L.A., Non-Noether constants of motion, J. Phys. A: Math. Gen., 1983, V.16, 1-7.
  2. Cariñena J.F., Marmo G., Rañada M.F., Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A: Math. Gen., 2002, V.35, L679-L686.
  3. Caseiro R., Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 2002, V.126, 617-630.
  4. Crampin M., Tangent bundle geometry for Lagrangian dynamics, J. Phys. A: Math. Gen., 1983, V.16, 3755-3772.
  5. Damianou P.A., Symmetries of Toda equations, J. Phys. A, 1993, V.26, 3791-3796.
  6. Damianou P.A., Sophocleous C., Lie point symmetries of Hamiltonian systems, Bull. Greek Math. Soc., 2000, V.44, 87-96.
  7. Fernandes R.L., On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A: Math. Gen., 1993, V.26, 3797-3803.
  8. Fokas A.S., Lagerstrom P.A., Quadratic and cubic invariants in classical mechanics, J. Math. Anal. Appl. 1980, V.74, 325-341.
  9. Gravel S., Hamiltonians separable in Cartesian coordinates and third-order integrals of motion, J. Math. Phys., 2004, V.45, 1003-1019.
  10. Gravel S., Winternitz P., Superintegrability with third-order integrals in quantum and classical mechanics, J. Math. Phys., 2002, V.43, 5902-5912.
  11. Hietarinta J., Direct methods for the search of the second invariant, Phys. Rep., 1987, V.147, 87-154.
  12. Holt C.R., Construction of new integrable Hamiltonians in two degrees of freedom, J. Math. Phys., 1982, V.23, 1037-46.
  13. McLenaghan R.G., Smirnov R.G., The D., Towards a classification of cubic integrals of motion, in Proceedings of the First International Workshop "Superintegrability in Classical and Quantum Systems" (September 16-21, 2002, Montreal), Editors P. Tempesta et al., CRM Proc. Lecture Notes, V.37, Providence, RI, Amer. Math. Soc., 2004, 199-209.
  14. McSween E., Winternitz P., Integrable and superintegrable Hamiltonian systems in magnetic fields, J. Math. Phys., 2000, V.41, 2957-2967.
  15. Prince G., Toward a classification of dynamical symmetries in classical mechanics, Bull. Austral. Math. Soc., 1983, V.27, 53-71.
  16. Rañada M.F., Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys., 1997, V.38, 4165-4178.
  17. Rañada M.F., Superintegrability of the Calogero-Moser system: constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys., 1999, V.40, 236-247.
  18. Rañada M.F., Dynamical symmetries, bi-Hamiltonian structures, and superintegrable n=2 systems, J. Math. Phys., 2000, V.41, 2121-2134.
  19. Rañada M.F., Santander M., Complex Euclidean super-integrable potentials, potentials of Drach, and potential of Holt, Phys. Lett. A, 2001, V.278, 271-279.
  20. Sergyeyev A., A simple way of making a Hamiltonian system into a bi-Hamiltonian one, Acta Appl. Math., 2004, V.83, 183-197.
  21. Sheftel M., On the classification of third-order integrals of motion in two-dimensional quantum mechanics, in Proceedings of the First International Workshop "Superintegrability in Classical and Quantum Systems" (September 16-21, 2002, Montreal), Editors P. Tempesta et al., CRM Proc. Lecture Notes, V.37, Providence, RI, Amer. Math. Soc., 2004, 187-197.
  22. Smirnov R.G., Bi-Hamiltonian formalism: a constructive approach, Lett. Math. Phys., 1997, V.41, 333-347.
  23. Thompson G., Polynomial constants of motion in flat space, J. Math. Phys., 1984, V.25, 3474-3478.

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