Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 8A (Classical Dynamics)

With $x=\ell \sin \theta$ and $y=-a{t}^2/2-\ell \cos \theta$, discard a total derivative from the kinetic energy to obtainThus $\ddot{\theta }+(g-a)\sin \theta /\ell =0$. If $a=g$, then $\ddot{\theta }=0$: the pendulum is weightless in the lift and rotates uniformly (or remains fixed).

The equilibria are $\theta =0,\pi$ modulo $2\pi$ unless $a=g$, when every angle is an equilibrium. Linearisation shows $heta=0$ stable for $a<g$ and $\theta =\pi$ stable for $a>g$; the other is unstable.