Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 8B / b / Solution

Put $ϵ=|E|>0$. On the energy surface,soEquality occurs at a turning point, so this is the smallest possible bound.

Using the natural collision continuation through the singular point, a full orbit runs between both turning points. HenceSet $q=x^2/ϵ$. ThenThereforeIf instead one identifies the singularity as a reflecting endpoint and regards one half-line as the orbit, the action is half this value; the following scaling is identical.

Adiabatic invariance keeps $m/|E|$ constant. Thus doubling $m$ doubles $|E|$:The energy becomes twice as negative.