Abstract: We study the dynamics of a family of mechanical systems which includes the pendulum, on small neighbourhoods of an elliptic equilibrium and for long intervals of time using the next term, after Galileo's observation, in the Taylor's expansion of the period map. We characterize the dynamical behaviour of such family through a renormalization scheme acting on the dynamics of this family of mechanical systems. The main theorem states that the asymptotic limit of this renormalization scheme is universal: it is the same for all the elements in the considered class of mechanical systems. As a consequence we obtain a universal asymptotic focal decomposition for this family of mechanical systems. Furthermore, we obtain that the asymptotic trajectories have a Hamiltonian character and compute the action of each element in this family of trajectories. This is joint work with C. A. A. de Carvalho, M. M. Peixoto and D. Pinheiro.