Equations of motion for a time-dependent open system: An algebraic approach

► Based on the concept of quantum densities an algebraic equation is introduced. ► Heisenberg equation and hypervirial theorem are derived based on the algebraic equation. ► Quantum Navier-Stokes equation is derived based on the algebraic equation. ► Differential form of the force law and local virial theorem are extracted. ► Central equations in QTAIM are extracted without referring to quantum stationary action.