Paper 2, Section II, B

Differential Equations | Part IA, 2014

Consider the damped pendulum equation

d2θdt2+cdθdt+sinθ=0\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}+\sin \theta=0

where cc is a positive constant. The energy EE, which is the sum of the kinetic energy and the potential energy, is defined by

E(t)=12(dθdt)2+1cosθE(t)=\frac{1}{2}\left(\frac{d \theta}{d t}\right)^{2}+1-\cos \theta

(i) Verify that E(t)E(t) is a decreasing function.

(ii) Assuming that θ\theta is sufficiently small, so that terms of order θ3\theta^{3} 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 cc.

(iii) By rewriting ()(*) as a system of equations for x(t)=θx(t)=\theta and y(t)=θ˙y(t)=\dot{\theta}, find all stationary points of ()(*) and discuss their nature for all cc, except c=2c=2.

(iv) Draw the phase plane curves for the particular case c=1c=1.

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