Consider the damped pendulum equation
where is a positive constant. The energy , which is the sum of the kinetic energy and the potential energy, is defined by
(i) Verify that is a decreasing function.
(ii) Assuming that is sufficiently small, so that terms of order can be neglected, find an approximation for the general solution of in terms of two arbitrary constants. Discuss the dependence of this approximate solution on .
(iii) By rewriting as a system of equations for and , find all stationary points of and discuss their nature for all , except .
(iv) Draw the phase plane curves for the particular case .