Solution
= Solution
Liouville transport of the phase-space density along the particle trajectories gives
$$
\frac{df}{d\eta}
=\frac{\partial f}{\partial\eta}
+\mathbf x'\mathbin\cdot\nabla_xf
+\mathbf p'\mathbin\cdot\nabla_pf=0.
$$
Using the equations of motion produces the <Collisionless dark-matter Vlasov equation>
$$
\boxed{\frac{\partial f}{\partial\eta}
+\frac{\mathbf p}{am}\mathbin\cdot\nabla_xf
-am\nabla\phi\mathbin\cdot\nabla_pf=0}.
$$