Matrix G-Strands

TL;DR

This paper explores the extension of integrable Euler--Poincaré ODEs to PDEs within the matrix G-Strand framework, demonstrating integrability and connections to rod theory for various Lie groups.

Contribution

It introduces matrix G-Strand examples for SO(3), SO(4), and SE(3), and establishes conditions for their complete integrability through zero curvature representations.

Findings

Matrix G-Strand equations for SO(3), SO(4), and SE(3) are integrable.

The equations recover the exact rod theory in the SE(3) case.

A zero curvature representation is found, confirming integrability.

Abstract

We discuss three examples in which one may extend integrable Euler--Poincar\'e ODEs to integrable Euler--Poincar\'e PDEs in the matrix G-Strand context. After describing matrix G-Strand examples for $SO(3)$ and $SO(4)$ we turn our attention to $SE(3)$ where the matrix G-Strand equations recover the exact rod theory in the convective representation. We then find a zero curvature representation (ZCR) of these equations and establish the conditions under which they are completely integrable. Thus, the G-Strand equations turn out to be a rich source of integrable systems. The treatment is meant to be expository and most concepts are explained in examples in the language of vectors in $\mathbb{R}^3$.