Phase-space divergence (${\nabla }_z\cdot b$)

The divergence of the vector field generating a flow on phase space. Its time integral determines the Jacobian determinant of the characteristic flow map: $d\log |\det D{\Phi }_t|/dt=({\nabla }_z\cdot b)({\Phi }_t,t)$. A Hamiltonian vector field has zero phase-space divergence in canonical coordinates.