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


SIGMA 9 (2013), 015, 13 pages      arXiv:1301.1432      https://doi.org/10.3842/SIGMA.2013.015

On a Trivial Family of Noncommutative Integrable Systems

Andrey V. Tsiganov
St. Petersburg State University, St. Petersburg, Russia

Received October 17, 2012, in final form February 18, 2013; Published online February 22, 2013

Abstract
We discuss trivial deformations of the canonical Poisson brackets associated with the Toda lattices, relativistic Toda lattices, Henon-Heiles, rational Calogero-Moser and Ruijsenaars-Schneider systems and apply one of these deformations to construct a new trivial family of noncommutative integrable systems.

Key words: bi-Hamiltonian geometry; noncommutative integrable systems.

pdf (321 kb)   tex (19 kb)

References

  1. Abraham R., Marsden J.E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Co. Inc., Reading, Mass., 1978.
  2. Arnol'd V.I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, Vol. 60, 2nd ed., Springer-Verlag, New York, 1989.
  3. Arsie A., Lorenzoni P., On bi-Hamiltonian deformations of exact pencils of hydrodynamic type, J. Phys. A: Math. Theor. 44 (2011), 225205, 31 pages, arXiv:1101.0167.
  4. Ayadi V., Fehér L., Görbe T.F., Superintegrability of rational Ruijsenaars-Schneider systems and their action-angle duals, J. Geom. Symmetry Phys. 27 (2012), 27-44, arXiv:1209.1314.
  5. Benenti S., Chanu C., Rastelli G., The super-separability of the three-body inverse-square Calogero system, J. Math. Phys. 41 (2000), 4654-4678.
  6. Broadbridge P., Chanu C.M., Miller Jr. W., Solutions of Helmholtz and Schrödinger equations with side condition and nonregular separation of variables, SIGMA 8 (2012), 089, 31 pages, arXiv:1209.2019.
  7. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2196.
  8. Das A., Okubo S., A systematic study of the Toda lattice, Ann. Physics 190 (1989), 215-232.
  9. Degiovanni L., Magri F., Sciacca V., On deformation of Poisson manifolds of hydrodynamic type, Comm. Math. Phys. 253 (2005), 1-24, nlin.SI/0103052.
  10. Fernandes R.L., On the master symmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A: Math. Gen. 26 (1993), 3797-3803.
  11. Gonera C., Nutku Y., Super-integrable Calogero-type systems admit maximal number of Poisson structures, Phys. Lett. A 285 (2001), 301-306, nlin.SI/0105056.
  12. Grammaticos B., Dorizzi B., Ramani A., Hamiltonians with high-order integrals and the "weak-Painlevé" concept, J. Math. Phys. 25 (1984), 3470-3473.
  13. Griffiths P., Harris J., Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons Inc., New York, 1994.
  14. Grigoryev Yu.A., Tsiganov A.V., On the Darboux-Nijenhuis variables for the open Toda lattice, SIGMA 2 (2006), 097, 15 pages, nlin.SI/0701004.
  15. Grigoryev Yu.A., Tsiganov A.V., Separation of variables for the generalized Henon-Heiles system and system with quartic potential, J. Phys. A: Math. Theor. 44 (2011), 255202, 9 pages, arXiv:1012.0468.
  16. Grigoryev Yu.A., Tsiganov A.V., Symbolic software for separation of variables in the Hamilton-Jacobi equation for the L-systems, Regul. Chaotic Dyn. 10 (2005), 413-422, nlin.SI/0505047.
  17. Ibort A., The geometry of dynamics, Extracta Math. 11 (1996), 80-105.
  18. Ibort A., Magri F., Marmo G., Bihamiltonian structures and Stäckel separability, J. Geom. Phys. 33 (2000), 210-228.
  19. Jacobi C.G.J., Vorlesungen über dynamik, G. Reimer, Berlin, 1884.
  20. Jost R., Poisson brackets (an unpedagogical lecture), Rev. Modern Phys. 36 (1964), 572-579.
  21. Khesin B., Tabachnikov S., Contact complete integrability, Regul. Chaotic Dyn. 15 (2010), 504-520, arXiv:0910.0375.
  22. Kuznetsov V.B., Tsiganov A.V., Separation of variables for the quantum relativistic Toda lattices, J. Math. Sci. 80 (1994), 1802-1810, hep-th/9402111.
  23. Lichnerowicz A., Les variétés de Poisson et leurs algèbres de Lie associées, J. Differential Geometry 12 (1977), 253-300.
  24. Maciejewski A.J., Przybylska M., Tsiganov A.V., On algebraic construction of certain integrable and super-integrable systems, Phys. D 240 (2011), 1426-1448, arXiv:1011.3249.
  25. Magri F., Casati P., Falqui G., Pedroni M., Eight lectures on integrable systems, in Integrability of Nonlinear Systems (Pondicherry, 1996), Lecture Notes in Phys., Vol. 495, Springer, Berlin, 1997, 256-296.
  26. Oevel W., Fuchssteiner B., Zhang H., Ragnisco O., Mastersymmetries, angle variables, and recursion operator of the relativistic Toda lattice, J. Math. Phys. 30 (1989), 2664-2670.
  27. Olver P.J., Rosenau P., Group-invariant solutions of differential equations, SIAM J. Appl. Math. 47 (1987), 263-278.
  28. Ovsiannikov L.V., Group analysis of differential equations, Academic Press Inc., New York, 1982.
  29. Suris Y.B., On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A 180 (1993), 419-429.
  30. Tempesta P., Tondo G., Generalized Lenard chains, separation of variables, and superintegrability, Phys. Rev. E 85 (2012), 046602, 11 pages, arXiv:1205.6937.
  31. Tsiganov A.V., On bi-integrable natural Hamiltonian systems on Riemannian manifolds, J. Nonlinear Math. Phys. 18 (2011), 245-268, arXiv:1006.3914.
  32. Tsiganov A.V., On natural Poisson bivectors on the sphere, J. Phys. A: Math. Theor. 44 (2011), 105203, 21 pages, arXiv:1010.3492.
  33. Tsiganov A.V., On the Poisson structures for the nonholonomic Chaplygin and Veselova problems, Regul. Chaotic Dyn. 17 (2012), 439-450.
  34. Tsiganov A.V., On two different bi-Hamiltonian structures for the Toda lattice, J. Phys. A: Math. Theor. 40 (2007), 6395-6406, nlin.SI/0701062.
  35. Waksjö C., Rauch-Wojciechowski S., How to find separation coordinates for the Hamilton-Jacobi equation: a criterion of separability for natural Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 301-348.


Previous article  Next article   Contents of Volume 9 (2013)