40 research outputs found
Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
We study the Euler-Lagrange equations for a parameter dependent -invariant
Lagrangian on a homogeneous -space. We consider the pullback of the
parameter dependent Lagrangian to the Lie group , emphasizing the special
invariance properties of the associated Euler-Poincar\'e equations with
advected parameters
The path group construction of Lie group extensions
We present an explicit realization of abelian extensions of infinite
dimensional Lie groups using abelian extensions of path groups, by generalizing
Mickelsson's approach to loop groups and the approach of
Losev-Moore-Nekrasov-Shatashvili to current groups. We apply our method to
coupled cocycles on current Lie algebras and to Lichnerowicz cocycles on the
Lie algebra of divergence free vector fields.Comment: 16 page
Geodesic Equations on Diffeomorphism Groups
We bring together those systems of hydrodynamical type that can be written as
geodesic equations on diffeomorphism groups or on extensions of diffeomorphism
groups with right invariant or metrics. We present their formal
derivation starting from Euler's equation, the first order equation satisfied
by the right logarithmic derivative of a geodesic in Lie groups with right
invariant metrics.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA