Matrix G-Strands

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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References in corpus (4)

• Multi-component generalizations of the {CH} equation: Geometrical Aspects, Peakons and Numerical Examples
• $G$-Strands
• Dynamics of charged molecular strands
• Relative Geodesics in the Special Euclidean Group

Cited by in corpus (6)

• Importance and effectiveness of representing the shapes of Cosserat rods and framed curves as paths in the special Euclidean algebra
• Simulation of viscoelastic Cosserat rods based on the geometrically exact dynamics of special Euclidean strands
• $G$-Strands on symmetric spaces
• Poisson-Poincaré reduction for Field Theories
• Integrable G-Strands on semisimple Lie groups
• Hamiltonian Reduction in Affine Principal Bundles