Bäcklund Transformations for the Kirchhoff Top

Symmetry Integrability and Geometry-Methods and Applications

article

Orlando Ragnisco¹  Federico Zullo¹ 

[1] Dipartimento di Fisica Universitá Roma Tre and Istituto Nazionale di Fisica Nucleare

关键词: Kirchhof f equations;    B¨acklund transformations;    integrable maps;    Lax representation;   

DOI  :  10.3842/SIGMA.2011.001

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown