Geodesic
On a Trivial Family of Noncommutative Integrable Systems
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: bi-Hamiltonian geometry; noncommutative integrable systems. 
@article{SIGMA_2013_9_a14,
     author = {Andrey V. Tsiganov},
     title = {On {a~Trivial} {Family} of {Noncommutative} {Integrable} {Systems}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a14/}
}
TY  - JOUR
AU  - Andrey V. Tsiganov
TI  - On a Trivial Family of Noncommutative Integrable Systems
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2013
VL  - 9
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a14/
LA  - en
ID  - SIGMA_2013_9_a14
ER  - 
%0 Journal Article
%A Andrey V. Tsiganov
%T On a Trivial Family of Noncommutative Integrable Systems
%J Symmetry, integrability and geometry: methods and applications
%D 2013
%V 9
%U http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a14/
%G en
%F SIGMA_2013_9_a14
Andrey V. Tsiganov. On a Trivial Family of Noncommutative Integrable Systems. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a14/

[1] Abraham R., Marsden J. E., Foundations of mechanics, 2nd ed., Benjamin/Cummings Publishing Co. Inc., Reading, Mass., 1978 | MR | Zbl

[2] Arnol'd V. I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60, 2nd ed., Springer-Verlag, New York, 1989 | DOI | MR

[3] Arsie A., Lorenzoni P., “On bi-Hamiltonian deformations of exact pencils of hydrodynamic type”, J. Phys. A: Math. Theor., 44 (2011), 225205, 31 pp., arXiv: 1101.0167 | DOI | MR | Zbl

[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 | DOI

[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 | DOI | MR | Zbl

[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 pp., arXiv: 1209.2019 | DOI

[7] Calogero F., “Solution of a three-body problem in one dimension”, J. Math. Phys., 10 (1969), 2191–2196 | DOI | MR

[8] Das A., Okubo S., “A systematic study of the Toda lattice”, Ann. Physics, 190 (1989), 215–232 | DOI | MR

[9] Degiovanni L., Magri F., Sciacca V., “On deformation of {P}oisson manifolds of hydrodynamic type”, Comm. Math. Phys., 253 (2005), 1–24, arXiv: nlin.SI/0103052 | DOI | MR | Zbl

[10] Fernandes R. L., “On the master symmetries and bi-{H}amiltonian structure of the Toda lattice”, J. Phys. A: Math. Gen., 26 (1993), 3797–3803 | DOI | MR | Zbl

[11] Gonera C., Nutku Y., “Super-integrable {C}alogero-type systems admit maximal number of Poisson structures”, Phys. Lett. A, 285 (2001), 301–306, arXiv: nlin.SI/0105056 | DOI | MR | Zbl

[12] Grammaticos B., Dorizzi B., Ramani A., “Hamiltonians with high-order integrals and the “weak-Painlevé” concept”, J. Math. Phys., 25 (1984), 3470–3473 | DOI | MR

[13] Griffiths P., Harris J., Principles of algebraic geometry, Wiley Classics Library, John Wiley Sons Inc., New York, 1994 | MR

[14] Grigoryev Yu. A., Tsiganov A. V., “On the Darboux–Nijenhuis variables for the open Toda lattice”, SIGMA, 2 (2006), 097, 15 pp., arXiv: nlin.SI/0701004 | DOI | MR | Zbl

[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 pp., arXiv: 1012.0468 | DOI | MR | Zbl

[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, arXiv: nlin.SI/0505047 | DOI | MR

[17] Ibort A., “The geometry of dynamics”, Extracta Math., 11 (1996), 80–105 | MR

[18] Ibort A., Magri F., Marmo G., “Bihamiltonian structures and Stäckel separability”, J. Geom. Phys., 33 (2000), 210–228 | DOI | MR | Zbl

[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 | DOI

[21] Khesin B., Tabachnikov S., “Contact complete integrability”, Regul. Chaotic Dyn., 15 (2010), 504–520, arXiv: 0910.0375 | DOI | MR | Zbl

[22] Kuznetsov V. B., Tsiganov A. V., “Separation of variables for the quantum relativistic Toda lattices”, J. Math. Sci., 80 (1994), 1802–1810, arXiv: hep-th/9402111 | DOI

[23] Lichnerowicz A., “Les variétés de {P}oisson et leurs algèbres de Lie associées”, J. Differential Geometry, 12 (1977), 253–300 | MR | Zbl

[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 | DOI | MR | Zbl

[25] Magri F., Casati P., Falqui G., Pedroni M., “Eight lectures on integrable systems”, Integrability of Nonlinear Systems (Pondicherry, 1996), Lecture Notes in Phys., 495, Springer, Berlin, 1997, 256–296 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[27] Olver P. J., Rosenau P., “Group-invariant solutions of differential equations”, SIAM J. Appl. Math., 47 (1987), 263–278 | DOI | MR | Zbl

[28] Ovsiannikov L. V., Group analysis of differential equations, Academic Press Inc., New York, 1982 | MR | Zbl

[29] Suris Y. B., “On the bi-Hamiltonian structure of Toda and relativistic Toda lattices”, Phys. Lett. A, 180 (1993), 419–429 | DOI | MR

[30] Tempesta P., Tondo G., “Generalized Lenard chains, separation of variables, and superintegrability”, Phys. Rev. E, 85 (2012), 046602, 11 pp., arXiv: 1205.6937 | DOI

[31] Tsiganov A. V., “On bi-integrable natural Hamiltonian systems on Riemannian manifolds”, J. Nonlinear Math. Phys., 18 (2011), 245–268, arXiv: 1006.3914 | DOI | MR | Zbl

[32] Tsiganov A. V., “On natural {P}oisson bivectors on the sphere”, J. Phys. A: Math. Theor., 44 (2011), 105203, 21 pp., arXiv: 1010.3492 | DOI | MR | Zbl

[33] Tsiganov A. V., “On the Poisson structures for the nonholonomic Chaplygin and Veselova problems”, Regul. Chaotic Dyn., 17 (2012), 439–450 | DOI | MR | Zbl

[34] Tsiganov A. V., “On two different bi-Hamiltonian structures for the Toda lattice”, J. Phys. A: Math. Theor., 40 (2007), 6395–6406, arXiv: nlin.SI/0701062 | DOI | MR | Zbl

[35] Waksjö C., Rauch-Wojciechowski S., “How to find separation coordinates for the {H}amilton–{J}acobi equation: a criterion of separability for natural {H}amiltonian systems”, Math. Phys. Anal. Geom., 6 (2003), 301–348 | DOI | MR | Zbl