 Solution

Phase space combines all independent canonical coordinates and momenta. Its Hamiltonian velocity has divergence ${\sum }_i[{\partial }_{q_i}{H}_{p_i}-{\partial }_{p_i}{H}_{q_i}]=0$ by equality of mixed derivatives. Thus an infinitesimal material phase volume has zero fractional rate of change. This is Liouville's theorem, valid also for explicitly time-dependent Hamiltonians. It preserves fine-grained volume, not necessarily the area of any projected coordinate pair.