Euler-Poincaré equations for $G$-Strands

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-Poincaré 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, ${\mathfrak{g}}^{\ast }$ 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.

To appear in Conference Proceedings for Physics and Mathematics of Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy) http://pmnp2013.dmf.unisalento.it/talks.shtml, 9 pages, no figures

References in corpus (1)

• $G$-Strands