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Backlund Transformations for the Kirchhoff Top

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 Added by Orlando Ragnisco
 Publication date 2010
  fields Physics
and research's language is English




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We construct Backlund 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.



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