Monday, May 20, 2013

1305.4010 (Darryl D. Holm et al.)

Matrix G-Strands    [PDF]

Darryl D. Holm, Rossen I. Ivanov
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 $\mathbb{R}^3$.
View original: http://arxiv.org/abs/1305.4010

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