Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)

Abstract

We show that the $m$-dimensional Euler--Manakov top on $s{o}^{\ast }(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\mathcal{V}}(k,m)$, and present its Lax representation with a rational parameter. We also describe an integrable two-valued symplectic map $\mathcal{B}$ on the 4-dimensional variety $\mathcal{V}(2,3)$. The map admits two different reductions, namely, to the Lie group SO(3) and to the coalgebra $s{o}^{\ast }(3)$. The first reduction provides a discretization of the motion of the classical Euler top in space and has a transparent geometric interpretation, which can be regarded as a discrete version of the celebrated Poinsot model of motion and which inherits some properties of another discrete system, the elliptic billiard. The reduction of $\mathcal{B}$ to $s{o}^{\ast }(3)$ gives a new explicit discretization of the Euler top in the angular momentum space, which preserves first integrals of the continuous system.

18 pages, 1 Figure

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