Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 8B (Classical Dynamics)

For a one-dimensional closed energy orbit, defineAfter locally inverting this relation to write $E=H(I)$, use the Hamilton--Jacobi generating functionand define the angle $\varphi =\partial W/\partial I$ modulo $2\pi$. The transformation is canonical, the Hamiltonian depends only on $I$, and

The adiabatic-invariance principle says that if an external parameter changes on a time scale much longer than the orbital period, and no separatrix is crossed, the action $I$ remains approximately constant. Its small within-cycle changes do not accumulate at leading adiabatic order.

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.