  Based on Koopman's theory of classical wavefunctions in phase space, we
present the Koopman-van Hove (KvH) formulation of classical mechanics as well
as some of its properties. In particular, we show how the associated classical
Liouville density arises as a momentum map associated to the unitary action of
strict contact transformations on classical wavefunctions. Upon applying the
Madelung transform from quantum hydrodynamics in the new context, we show how
the Koopman wavefunction picture is insufficient to reproduce arbitrary
classical distributions. However, this problem is entirely overcome by
resorting to von Neumann operators. Indeed, we show that the latter also allow
for singular $\delta-$like profiles of the Liouville density, thereby
reproducing point particles in phase space.
