B\"acklund Transformations for the Kirchhoff Top

TL;DR

This paper develops Bäcklund transformations for the Kirchhoff top, creating integrable, structure-preserving maps that discretize the system exactly and can sometimes be explicitly integrated, advancing understanding of discrete integrable systems.

Contribution

It introduces Bäcklund transformations for the Kirchhoff top based on its algebraic Poisson structure, providing exact discretizations and explicit solutions in special cases.

Findings

BTs preserve the Poisson structure and invariants.

In some cases, the maps are explicitly integrable.

The maps are conjectured to interpolate Hamiltonian flows.

Abstract

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.