2 research outputs found
Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations
Given a Lie-Poisson completely integrable bi-Hamiltonian system on
, we present a method which allows us to construct, under certain
conditions, a completely integrable bi-Hamiltonian deformation of the initial
Lie-Poisson system on a non-abelian Poisson-Lie group of dimension
, where is the deformation parameter. Moreover, we
show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on
that underly the dynamics of the deformed system and by making use of
the group law on , one may obtain two completely integrable Hamiltonian
systems on . By construction, both systems admit
reduction, via the multiplication in , to the deformed bi-Hamiltonian
system in . The previous approach is applied to two relevant
Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler
top systems.Comment: 23 pages, 2 figures. Revised versio