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The G-strand equations for a map R×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):R×R→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 g and its co-algebra, 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.
D. D. Holm, R. I. Ivanov, Euler-Poincaré equations for G-Strands, Physics and Mathematics of Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy) ; Journal of Physics: Conference Series 482 (2014) 012018 doi:10.1088/1742-6596/482/1/012018