**Exercise 6.4.** **Periodically driven pendulum**

**a**. Work out the details of the perturbation theory for the
primary driven pendulum resonances, as displayed in
figure 6.10.

**b**. Work out the details of the perturbation theory for the
stability of the inverted vertical equilibrium. Derive the resonance
Hamiltonian and plot its contours. Compare these contours to
surfaces of section for a variety of parameters.

**c**. Carry out the linear stability analysis leading to
equation (6.87). What is
happening in the upper part of figure 6.15?
Why is the system unstable when
criterion (6.87) predicts
stability? Use surfaces of section to investigate this parameter
regime.

**Exercise 6.5.** **Spin-orbit coupling**

A Hamiltonian for the spin-orbit problem described in
section 2.11.2 is

where the ignored terms are higher order in eccentricity *e*.

**a**. Find the widths and centers of the three primary resonances.
Compare the predictions for the widths to the island widths seen on
surfaces of section.
Write the criterion for resonance overlap and compare to numerical
experiments for the transition to large-scale chaos.

**b**. The fixed point of the synchronous island is offset from the
average rate of rotation. This is indicative of a ``forced''
oscillation of the rotation of the Moon. Develop a perturbative theory
for motion in the synchronous island by using a Lie transform to
eliminate the two non-synchronous resonances. Predict the location of
the fixed point at the center of the synchronous resonance on the
surface of section, and thus predict the amplitude of the forced
oscillation of the Moon.