Separation of variables for some systems with a fourth-order integral of motion

Abstract

ДЛЯ НЕКОТОРЫХ СИСТЕМ С ИНТЕГРАЛОМ ДВИЖЕНИЯ ЧЕТВЕРТОЙ СТЕПЕНИПостроены переменные разделения для интегрируемых деформаций Яхьи в случае волчка Ковалевской и системы Чаплыгина на сфере. В общем случае соответствующие квадратуры представляют собой отображение Абеля-Якоби на двумерном подмногообразии якобиана алгебраической кривой рода три, ко-торая не является гиперэллиптической.Ключевые слова: бигамильтонова геометрия, разделение переменных, волчок Кова-левской.

Simultaneous separation for the Neumann and Chaplygin systems

“…The separated relation (4.5) was found in [12] using the direct bi-Hamiltonian method discussed in the previous Section. Substituting L(λ) into the linear r-matrix equation (3.11) with r-matrices (3.16) or (3.17) we can recover the corresponding Poisson bivectors P ′ 1 and P ′ 2 .…”

“…Then we identify canonical transformation of the elliptic coordinates to the Chaplygin variables with the special Bäcklund transformation, which can be easily described using 2×2 Lax representation for the Neumann system. It allows us to obtain similar Bäcklund transformations for other curvilinear coordinate systems and to prove that known variables of separation for the system with quartic potential [28], for the Hénon-Heiles system [26] and for the Kowalevski top [12,35] are the standard curvilinear coordinates (elliptic, parabolic) after the Bäcklund transformations.…”

“…As above the separated relation (4.22) defines genus 3 non-hyperelliptic curve. These v-variables and separated relation (4.22) were obtained in [12,35] in the framework of bi-Hamiltonian geometry. Substituting L(λ) into the linear r-matrix equation (3.11) with r-matrices (3.16) or (3.17) we can recover the corresponding Poisson bivectors P ′ 1 and P ′ 2 compatible with the canonical bivector P in T * S 2 .…”

“…Roughly speaking, after the similarity transformation (3.6) zeroes of the two off-diagonal matrix elementsL 12 (λ = u i ) = 0 andL 21 (λ = v i ) = 0 form two families of separation variables with different properties. Below we want to discuss other differences between Lax matrices L(λ) (3.4) andL(λ) (3.6).…”