Document Type
Conference Paper
Rights
This item is available under a Creative Commons License for non-commercial use only
Disciplines
Applied mathematics
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.
DOI
https://doi.org/doi:10.1088/1742-6596/482/1/012018
Recommended Citation
D. D. Holm, & R. I. Ivanov. (2013). Euler-Poincaré equations for G-Strands. Physics and Mathematics of Nonlinear Phenomena, Gallipoli (Italy), 22-29 June 2013. Journal of Physics: Conference Series 482(1), 012018. doi:10.1088/1742-6596/482/1/012018
Funder
SFI
Included in
Dynamical Systems Commons, Dynamic Systems Commons, Non-linear Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons