  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.
