V. V. Sokolov, A. V. Tsiganov, “Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models”, TMF, 133:3 (2002), 485–500; Theoret. and Math. Phys., 133:3 (2002), 1730

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Teoreticheskaya i Matematicheskaya Fizika, 2002, Volume 133, Number 3, Pages 485–500
DOI: https://doi.org/10.4213/tmf413 (Mi tmf413)

This article is cited in 15 scientific papers (total in 15 papers)

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

V. V. Sokolov^a, A. V. Tsiganov^b

^a Landau Institute for Theoretical Physics, Centre for Non-linear Studies
^b St. Petersburg State University, Faculty of Physics

Full-text PDF (274 kB) Citations (15)

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DOI: https://doi.org/10.4213/tmf413

Abstract: We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.

Keywords: finite-dimensional integrable systems, Lax representation,

r

$r$ -matrix algebras, separation of variables.

English version:
Theoretical and Mathematical Physics, 2002, Volume 133, Issue 3, Pages 1730–1743
DOI: https://doi.org/10.1023/A:1021326727968

Bibliographic databases:

Language: Russian

Citation: V. V. Sokolov, A. V. Tsiganov, “Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models”, TMF, 133:3 (2002), 485–500; Theoret. and Math. Phys., 133:3 (2002), 1730–1743

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf413

https://doi.org/10.4213/tmf413

https://www.mathnet.ru/eng/tmf/v133/i3/p485

This publication is cited in the following 15 articles:

Borisov A. Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
Elmandouh A.A., “New Integrable Problems in a Rigid Body Dynamics With Cubic Integral in Velocities”, Results Phys., 8 (2018), 559–568
A. V. Tsiganov, “Bäcklund transformations for the Jacobi system on an ellipsoid”, Theoret. and Math. Phys., 192:3 (2017), 1350–1364
Yehia H.M. Elmandouh A.A., “Integrable 2D Time-Irreversible Systems with a Cubic Second Integral”, Adv. Math. Phys., 2016, 8958747
A. V. Tsyganov, “Razdelenie peremennykh dlya odnogo obobscheniya sistemy Chaplygina na sfere”, Nelineinaya dinam., 11:1 (2015), 179–185
A. P. Sozonov, A. V. Tsiganov, “Bäcklund transformations relating different Hamilton–Jacobi equations”, Theoret. and Math. Phys., 183:3 (2015), 768–781
A.V. Tsiganov, “On the Chaplygin system on the sphere with velocity dependent potential”, Journal of Geometry and Physics, 92 (2015), 94
Tsiganov, AV, “On maximally superintegrable systems”, Regular & Chaotic Dynamics, 13:3 (2008), 178
Tsiganov, AV, “The Poisson bracket compatible with the classical reflection equation algebra”, Regular & Chaotic Dynamics, 13:3 (2008), 191
Kostko, AL, “On the bi-Hamiltonian structures for the Goryachev-Chaplygin top”, Regular & Chaotic Dynamics, 13:1 (2008), 38
A. V. Tsiganov, “Darboux–Nijenhuis variables for open generalized Toda chains”, Theoret. and Math. Phys., 152:3 (2007), 1243–1257
Sokolov, VV, “Integrable quadratic classical Hamiltonians on so(4) and so(3,1)”, Journal of Physics A-Mathematical and General, 39:8 (2006), 1915
A. V. Tsiganov, “Toda Chains in the Jacobi Method”, Theoret. and Math. Phys., 139:2 (2004), 636–653
Borisov AV, Mamaev IS, “Necessary and sufficient conditions for the polynomial integrability of generalized Toda chains”, Doklady Mathematics, 69:1 (2004), 131–135
A. V. Tsiganov, “Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat”, Theoret. and Math. Phys., 135:2 (2003), 651–658

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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