Euler equations on the general linear group, cubic curves, and inscribed hexagons

TL;DR

This paper explores integrable Euler equations on the Lie algebra gl(3,R) by linking their evolution to inscribed hexagons on real cubic curves, revealing geometric insights into their integrability.

Contribution

It introduces a novel geometric interpretation of Euler equations as evolutions of inscribed hexagons on cubic curves, connecting algebraic and geometric perspectives.

Findings

Establishes a correspondence between Euler equations and inscribed hexagons.

Provides a geometric framework for understanding integrability.

Links algebraic equations to cubic curve geometry.

Abstract

We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.