Paper 2, Section II, B

Differential Equations | Part IA, 2015

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

d2θdt2+cdθdtdθdt+sinθ=0\frac{d^{2} \theta}{d t^{2}}+c \frac{d \theta}{d t}\left|\frac{d \theta}{d t}\right|+\sin \theta=0

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

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

By considering the cases of ω>0\omega>0 and ω<0\omega<0 separately, find explicit expressions for ω2\omega^{2} as a function of θ\theta. Discuss how the second term in ()(*) affects the nature of the equilibrium points.

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