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

TL;DR

This paper explores how reducible Abelian varieties, specifically those arising from Euler's two fixed centres problem, can be used to construct Lax matrices, linking algebraic geometry with integrable Hamiltonian systems.

Contribution

It demonstrates the construction of Lax matrices on factors of reducible Abelian varieties related to Euler's problem, extending the algebraic-geometric approach to integrable systems.

Findings

Lax matrices can be constructed on factors of reducible Abelian varieties.

Euler's two fixed centres problem relates to a product of elliptic curves.

The approach links algebraic geometry with integrable Hamiltonian systems.

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 $A_1,\ldots,A_k$. 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 $E_\pm$.