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arXiv · 1311.2126

Euler-Poincar\'e equations for $G$-Strands

Darryl D. Holm·Rossen I. Ivanov
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Abstract

The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The $G$-strand itself is the map $g(t,s): \mathbb{R}\times \mathbb{R}\to G$, where $t$ and $s$ are the independent variables of the $G$-strand equations. The Euler-Poincar\'e reduction of the variational principle leads to a formulation where the dependent variables of the $G$-strand equations take values in the corresponding Lie algebra $\mathfrak{g}$ and its co-algebra, $\mathfrak{g}^*$ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of different $G$-strand constructions, including matrix Lie groups and diffeomorphism group. In some cases the $G$-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.

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Darryl D. Holm, Rossen I. Ivanov. 2013-11-09. Euler-Poincar\'e equations for $G$-Strands. https://doi.org/10.1088/1742-6596/482/1/012018

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