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Figure 8. Pendulum of variable length, swinging through the angle (in a verticalplane).

small movement of the torso) causes the centre of gravity of the body to be raised and
lowered periodically. This can be modelled by treating the swing as a pendulum which
changes its length, by a small amount; the model equation for this (in the absence
of damping) is

for the angle of the swing, given see figure 8. We choose to represent the
a
child’s movement on the swing by where is (positive) constant
and is a constant frequency to be selected. We will further simplify the problem by
analysing only the initial stages of the motion when is small, so we write sin
(For larger amplitudes, we must retain sin this complicates the issue somewhat.
A number of more general observations about this problem can be found in Holmes,
1995.)
Thus we approximate equation (5.7) as

which we will solve using the method of multiple scales (cf. E4.1).
We take the fast scale as T = t (or, more generally, but there is no advantage
in this, for we are led to the choice and, by virtue of the term in a slow
scale thus we have the identity