 Rotating-frame collisionless Boltzmann equation

In physical polar velocity components, a planar rotating system has accelerations ${\dot{v}}_R=2{\Omega }_b{v}_{\varphi }+v_{\varphi }^2/R+{\psi }_{\mathrm{eff},R}$ and ${\dot{v}}_{\varphi }=-2{\Omega }_b{v}_R-v_R{v}_{\varphi }/R+{\psi }_{\mathrm{eff},\varphi }/R$. The Collisionless Boltzmann equation is constant distribution along these characteristics. Its physical phase-space measure includes the spatial Jacobian $R$.