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 (qCG) system, a q-Poincare Gaudin system and a system of Ruijsenaars type arising from the same (non-coboundary) q-deformation of the (1 + 1) Poincare algebra. While the complete integrability of all these systems was already well known, being in fact encoded in their construction, no explicit solution was available until now. In particular, it turns out that there exists an open subset of the whole phase space where the orbits of the qCG system are periodic with the same period. Also, a unified interpretation of all these systems as different Poisson-Lie dynamics on the same three-dimensional solvable Lie group is given.