Euler-Poincaré equations for G-Strands

Darryl D. Holm, Rossen I. Ivanov

Research output: Contribution to journalConference articlepeer-review

Abstract

The G-strand equations for a map × 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) : × → 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 and its co-algebra, * 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.

Original languageEnglish
Article number012018
JournalJournal of Physics: Conference Series
Volume482
Issue number1
DOIs
Publication statusPublished - 2014
EventPhysics and Mathematics of Nonlinear Phenomena 2013, PMNP 2013 - Gallipoli, Italy
Duration: 22 Jun 201329 Jun 2013

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