  Galileo, in the XVII century, observed that the small oscillations of a
pendulum seem to have constant period. In fact, the Taylor expansion of the
period map of the pendulum is constant up to second order in the initial
angular velocity around the stable equilibrium. It is well known that, for
small oscillations of the pendulum and small intervals of time, the dynamics of
the pendulum can be approximated by the dynamics of the harmonic oscillator. We
study the dynamics of a family of mechanical systems that includes the pendulum
at small neighbourhoods of the equilibrium but after long intervals of time so
that the second order term of the period map can no longer be neglected. We
analyze such dynamical behaviour 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 an universal asymptotic focal decomposition for this
family of mechanical systems. This paper is intended to be the first of a
series of articles aiming at a semiclassical quantization of systems of the
pendulum type as a natural application of the focal decomposition associated to
the two-point boundary value problem.
