Invariant variational problems on homogeneous spaces

Abstract

a b s t r a c tCovariant Euler-Poincaré equations on homogeneous spaces are studied, including the special case of strands on homogeneous spaces. Space-time strands on homogeneous spaces are treated also dynamically, using affine Euler-Poincaré reduction.

Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

“…It follows that the corresponding EP equations (4.7) for spin systems are invariant under the action (4.9), hence (4.7) can be seen as an equation on C ∞ (M, G)/G. This setting of affine EP reduction is used in [12] for the dynamical description of spacetime strands on homogeneous spaces. Covariant EP equations on homogeneous spaces provide another frame to describe the dynamics of space-time strands on homogeneous spaces.…”

G -Strands on symmetric spaces

“…Integrable systems on symmetric spaces of finite dimensional Lie algebras have been well studied in the literature, see [AF87,FK83,For84,GG10,GGK05]. A more general but similar construction would be on homogeneous spaces, as studied recently in [Viz15a,Viz15b] also in the context of reduction by symmetry. This construction is different from the semidirect product G-strands explored in [HI14] for the special Euclidean group SE(3) := SO(3) R 3 with Lie algebra commutation relations [so(3), so(3)] ⊂ so(3) and [so(3), R 3 ] ⊂ R 3 , because for symmetric spaces, an additional commutation relation occurs involving the complementary space p…”

“…17) for an arbitrary complex spectral parameter λ. This equivalent formulation of the equation(2.15) makes this model integrable by the means of the inverse scattering method.…”