Past exam of the kinetic theory course of the University of Oxford / 2024 / 3 / c / Solution

Use the stated Fourier series normalization, so $g(\mathbf{\theta },\mathbf{J},t)={\sum }_{\mathbf{n}}{g}_{\mathbf{n}}(\mathbf{J},t)e^{i\mathbf{n}\cdot \mathbf{\theta }}$. Then b givesFor the Laplace transform with kernel $e^{i\omega t}$, integration by parts yieldsThe upper boundary vanishes for sufficiently large $\mathrm{Im}\omega >0$, and the initial term is zero by hypothesis. ThusorThe denominator identifies the orbital resonance between the perturbation and unperturbed orbital frequencies. The zero harmonic has no driving term and remains zero with these initial data.