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Abstract

We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra e(3). Using this map we establish equivalence of the Steklov-Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra so(4) we use the twisted Poisson map between the bi-Hamiltonian manifolds e(3) and so(4).

New Variables of Separation for the Steklov-Lyapunov System

“…Here we want to add some new details in the known coincidence of the Kötter variables of separation v 1,2 with the elliptic coordinates on the sphere. According to [23,25] there is a Poisson map, which identifies the Steklov-Lyapunov system with a system that describes motion on the surface of a unit two-dimensional sphere S 2 in a fourth-degree polynomial potential field. This dynamical system is separable in standard elliptic coordinates on the sphere, and the inverse Poisson map allows us to get complete solution of the Steklov-Lyapunov system.…”

Compatible Lie-Poisson brackets on the Lie algebras e(3) and so(4)

“…If we set (4.9). It can be assumed that precisely this fact implies the existence of two "natural" 2×2 Lax representations for this system: with an elliptic dependence or with a rational dependence on the spectral parameter [11], [17].…”

Согласованные скобки Ли - Пуассона на алгебрах Ли $e(3)$ и $so(4)$

“…Таким образом, система Стеклова-Ляпунова может быть связана с двумя различными пучками скобок Ли-Пуассона (4.7) и (4.9). Можно предположить, что именно этот факт приводит к существованию двух "естественных" (2 × 2)-представлений Лакса для этой интегрируемой системы -с эллиптической зависимостью и с рациональной зависимостью от спектрального параметра [11], [17].…”