Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 313 / 1 / iv / Solution

Use the four-parameter ansatz from part (ii), now allowing every parameter to vary slowly. Oddness of $y{f}_E(y{)}^2$ and the stationary-profile integrals give the collective-coordinate effective LagrangianwhereVariation of $\theta$ makes $N(E)$, and hence $E$, constant. Variation of $u$, $X$, and $E$ then givesThus the center follows the classical Hamiltonianto leading adiabatic order. The first neglected profile correction is of order ${ϵ}^3$, because the term linear in $y$ integrates to zero.