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

This article is cited in 7 scientific papers (total in 7 papers)

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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Full text: http://emis.mi.ras.ru/journals/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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\by Orlando Ragnisco, Federico Zullo
\paper B\"acklund Transformations for the Kirchhoff Top
\jour SIGMA
\yr 2011
\vol 7
\papernumber 001
\totalpages 13
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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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. O. Ragnisco, F. Zullo, “Quantum Bäcklund transformations: Some ideas and examples”, Theoret. and Math. Phys., 172:2 (2012), 1160–1171  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    3. Zullo F., “Backlund Transformations and Hamiltonian Flows”, J. Phys. A-Math. Theor., 46:14 (2013), 145203  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Zhou Ru-Guang, “A Backlund Transformation of the Restricted Mkdv Flow with a Rosochatius Deformation”, Commun. Theor. Phys., 60:3 (2013), 263–265  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Zullo F., “On an Integrable Discretisation of the Ablowitz-Ladik Hierarchy”, J. Math. Phys., 54:5 (2013), 053515  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Zullo F., “a Q-Difference Baxter Operator For the Ablowitz-Ladik Chain”, J. Phys. A-Math. Theor., 48:12 (2015), 125205  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. A. V. Tsiganov, “Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations”, Proc. Steklov Inst. Math., 295 (2016), 243–273  mathnet  crossref  crossref  mathscinet  isi  elib
  • Symmetry, Integrability and Geometry: Methods and Applications
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