Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2010-07-15
SIGMA 7 (2011), 001, 13 pages
Nonlinear Sciences
Exactly Solvable and Integrable Systems
Scientific paper
10.3842/SIGMA.2011.001
We construct B\"acklund 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.
Ragnisco Orlando
Zullo Federico
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