Paper 2, Section II, B

Write as a system of two first-order equations the second-order equation

$\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0$

where $c$ is a small, positive constant, and find its equilibrium points. What is the nature of these points?

Draw the trajectories in the $(\theta, \omega)$ plane, where $\omega=d \theta / d t$, in the neighbourhood of two typical equilibrium points.

By considering the cases of $\omega>0$ and $\omega<0$ separately, find explicit expressions for $\omega^{2}$ as a function of $\theta$. Discuss how the second term in $(*)$ affects the nature of the equilibrium points.

*Typos? Please submit corrections to this page on GitHub.*