  This paper extends the Madelung-Bohm formulation of quantum mechanics to
describe the time-reversible interaction of classical and quantum systems. The
symplectic geometry of the Madelung transform leads to identifying hybrid
classical-quantum Lagrangian paths extending the Bohmian trajectories from
standard quantum theory. As the classical symplectic form is no longer
preserved, the nontrivial evolution of the Poincar\'e integral is presented
explicitly. Nevertheless, the classical phase-space components of the hybrid
Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum
coordinate and this flow is associated to the motion of the classical
subsystem. In addition, the continuity equation of the joint classical-quantum
density is presented explicitly. While the von Neumann density operator of the
quantum subsystem is always positive-definite by construction, the hybrid
density is generally allowed to be unsigned. However, the paper concludes by
presenting an infinite family of hybrid Hamiltonians whose corresponding
evolution preserves the sign of the probability density for the classical
subsystem.
