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

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.