Document Type

Conference Paper

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

Applied mathematics

Publication Details

To appear in Conference Proceedings for Physics and Mathematics of Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy) in Journal of Physics: Conference Series (JPCS)

Abstract

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.

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