Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions

A general procedure to get the explicit solution of the equations of motion for N-body classical Hamiltonian systems equipped with coalgebra symmetry is introduced by defining a set of appropriate collective variables which are based on the iterations of the coproduct map on the generators of the algebra. In this way several examples of N-body dynamical systems obtained from q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2) Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of Ruijsenaars type arising from the same (non co-boundary) q-deformation of the (1+1) Poincare' algebra. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three dimensional solvable Lie group is given.