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 SIGMA, 2011, Volume 7, 001, 13 pages (Mi sigma559)

Bäcklund Transformations for the Kirchhoff Top

Orlando Ragniscoab, Federico Zulloab

a Dipartimento di Fisica Universitá Roma Tre
b Istituto Nazionale di Fisica Nucleare, Sezione di Roma, I-00146 Roma, Italy

Abstract: We construct Bäcklund transformations (BTs) for the Kirchhoff top by taking advantage of the common algebraic Poisson structure between this system and the $sl(2)$ trigonometric Gaudin model. Our BTs are integrable maps providing an exact time-discretization of the system, inasmuch as they preserve both its Poisson structure and its invariants. Moreover, in some special cases we are able to show that these maps can be explicitly integrated in terms of the initial conditions and of the “iteration time” $n$. Encouraged by these partial results we make the conjecture that the maps are interpolated by a specific one-parameter family of hamiltonian flows, and present the corresponding solution. We enclose a few pictures where the orbits of the continuous and of the discrete flow are depicted.

Keywords: Kirchhoff equations; Bäcklund transformations; integrable maps; Lax representation

DOI: https://doi.org/10.3842/SIGMA.2011.001

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Bibliographic databases:

ArXiv: 1007.2607
Document Type: Article
MSC: 37J35; 70H06; 70H15
Received: July 20, 2010; in final form December 14, 2010; Published online January 3, 2011
Language: English

Citation: Orlando Ragnisco, Federico Zullo, “Bäcklund Transformations for the Kirchhoff Top”, SIGMA, 7 (2011), 001, 13 pp.

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Zullo F., “Backlund transformations for the elliptic Gaudin model and a Clebsch system”, J Math Phys, 52:7 (2011), 073507
2. O. Ragnisco, F. Zullo, “Quantum Bäcklund transformations: Some ideas and examples”, Theoret. and Math. Phys., 172:2 (2012), 1160–1171
3. Zullo F., “Backlund Transformations and Hamiltonian Flows”, J. Phys. A-Math. Theor., 46:14 (2013), 145203
4. Zhou Ru-Guang, “A Backlund Transformation of the Restricted Mkdv Flow with a Rosochatius Deformation”, Commun. Theor. Phys., 60:3 (2013), 263–265
5. Zullo F., “On an Integrable Discretisation of the Ablowitz-Ladik Hierarchy”, J. Math. Phys., 54:5 (2013), 053515
6. Zullo F., “a Q-Difference Baxter Operator For the Ablowitz-Ladik Chain”, J. Phys. A-Math. Theor., 48:12 (2015), 125205
7. A. V. Tsiganov, “Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations”, Proc. Steklov Inst. Math., 295 (2016), 243–273
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