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Bäcklund Transformations for the Kirchhoff Top
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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Bäcklund Transformations for the Kirchhoff Top
Інший ID:
2010 Mathematics Subject Classification: 37J35; 70H06; 70H15
DOI:10.3842/SIGMA.2011.001
DOI:10.3842/SIGMA.2011.001
Посилання:
Bäcklund Transformations for the Kirchhoff Top / O. Ragnisco, F. Zullo // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 13 назв. — англ.
Підтримка:
This paper is a contribution to the Proceedings of the Conference “Integrable Systems and Geometry” (August 12–17, 2010, Pondicherry University, Puducherry, India). The full collection is available at http://www.emis.de/journals/SIGMA/ISG2010.html.
We are grateful to both referees for their constructive comments and criticisms, and in particular to one of them for his crucial remarks and for having brought to our attention the article [10]. The research underlying this paper has been partially supported by the Italian MIUR, Research Project “Integrable Nonlinear Evolutions, continuous and discrete: from Water Waves downwards to Symplectic Map”, Prot. n. 20082K9KXZ/005, in the framework of the PRIN 2008: “Geometrical Methods in the Theory of Nonlinear Waves and Applications”.
Дата:
2011
Переглядів:
746
Завантажень:
287
Анотація:
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.
