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arXiv · 1305.4010

Matrix G-Strands

Darryl D. Holm·Rossen I. Ivanov
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Abstract

We discuss three examples in which one may extend integrable Euler--Poincaré ODEs to integrable Euler--Poincaré 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 $\mathbb{R}^3$.

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Darryl D. Holm, Rossen I. Ivanov. 2014-02-09. Matrix G-Strands. https://doi.org/10.1088/0951-7715%2F27%2F6%2F1445

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