Reducible Abelian varieties and Lax matrices for Euler's problem of two fixed centres

Abstract

Abel's quadratures for integrable Hamiltonian systems are defined up to a group law of the corresponding Abelian variety $A$. If $A$ is isogenous to a direct product of Abelian varieties $A\cong A_1\times \cdots \times A_k$, the group law can be used to construct various Lax matrices on the factors $A_1,\dots ,A_k$. As an example, we discuss 2-dimensional reducible Abelian variety $A=E_{+}\times E_{-}$, which is a product of 1-dimensional varieties $E_{\pm }$ obtained by Euler in his study of the two fixed centres problem, and the Lax matrices on the factors $E_{\pm }$.

arXiv comment13 pages, 2 figures, AMS fonts

References in corpus (14)

• Separation of Variables. New Trends.
• Linear $r$-matrix algebra for classical separable systems
• Backlund transformations for finite-dimensional integrable systems: a geometric approach
• Simultaneous separation for the Neumann and Chaplygin systems
• On auto and hetero Backlund transformations for the Henon-Heiles systems
• Integrable flows and Backlund transformations on extended Stiefel varieties with application to the Euler top on the Lie group SO(3)
• Lax representation for two--particle dynamics splitting on two tori
• Superintegrable systems and Riemann-Roch theorem
• The Clebsch System
• A complete and explicit solution to the three-dimensional problem of two fixed centres
• The Kepler problem: polynomial algebra of non-polynomial first integrals
• Scattering invariants in Euler's two-center problem
• Isogenous components of Jacobian surfaces
• Reduction of divisors for classical superintegrable $GL(3)$ magnetic chain

Cited by in corpus (2)

• Reduction of divisors and Clebsch system
• Reduction of divisors and Kowalevski top