 Solution

ID: past-exam-of-the-kinetic-theory-course-of-the-university-of-oxford/2024/3/b/solution

Since

H_{0}

and

f_{0}

depend only on the action variables,

{f_{0}, H_{0}} = 0

. Drop the quadratic term

{δ f, δ Φ^{tot}}

from the Vlasov equation in a. Defining the orbital frequency vector by

Ω (J) = \partial H_{0} / \partial J

\partial_{t} δ f + Ω \cdot \partial_{θ} δ f - \frac{\partial f _{0}}{\partial J} \cdot \partial_{θ} δ Φ^{tot} = 0.

The sign of the driving term follows from the second term in the canonical Poisson bracket, because the background has no angle dependence.

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