Rotating-frame collisionless Boltzmann equation ($dF/dt=0$)

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 $R$.

The zeroth velocity moment gives ${\partial }_t\rho +R^{-1}{\partial }_R(R\rho ⟨v_R⟩)+R^{-1}{\partial }_{\varphi }(\rho ⟨v_{\varphi }⟩)=0$. In the rotating-frame collisionless Boltzmann equation, velocity integration by parts supplies the geometric $\rho ⟨v_R⟩/R$ term. This equation expresses conservation of planar mass.