Euler-Poincar\'e equations for $G$-Strands

Publication date: 8 November 2013

Abstract: The

G

-strand equations for a map

m a t h b b R i m e s m a t h b b 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) : m a t h b b R i m e s m a t h b b R o 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

m a t h f r a k g

and its co-algebra,

m a t h f r a k 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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