Equivalence of the pumping of a swing and the parametric resonance

In a model of a swing where the swing is regarded as a pendulum of variable length under friction, we analyse the mechanical equivalence of the pumping manner of the swing. It is shown from the similarity between the forms of equations that for some parameter region the pumping of the swing is nearly equivalent to an external force exerted by an external agent vibrating the axis of another pendulum vertically. In addition, we discuss the parametric resonance of the swing in terms of regular perturbation theory. It is observed that regular perturbation theory can be a good tool for finding the resonance frequency in spite of its weakness in long-term prediction