Cylindrical continuity equation for a rotating stellar system (${\rho }_t+R^{-1}(R\rho u_R{)}_R+R^{-1}(\rho u_{\varphi }{)}_{\varphi }=0$)

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.