Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 001, 13 pages      arXiv:1007.2607      http://dx.doi.org/10.3842/SIGMA.2011.001
Contribution to the Proceedings of the Conference “Integrable Systems and Geometry”

Bäcklund Transformations for the Kirchhoff Top

Orlando Ragnisco and Federico Zullo
Dipartimento di Fisica Universitá Roma Tre and Istituto Nazionale di Fisica Nucleare, Sezione di Roma, I-00146 Roma, Italy

Received July 20, 2010, in final form December 14, 2010; Published online January 03, 2011

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.

Key words: Kirchhoff equations; Bäcklund transformations; integrable maps; Lax representation.

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