  A recent formalism capturing the classical-quantum coupling in a Hamiltonian
theory for probabilistic classical mechanics has been proposed: the Koopman-van
Hove formulation. The aims of this report are twofolds. First, we rigourously
expose this new framework for a non-expert audience and present a construction
for the operatorial analogues to the classical angular and linear momentum
operators. We then investigate the group actions generating their average as
momentum maps as well as their associated dynamics. Finally, we illustrate the
theory by deriving the Koopman-van Hove formulation to the Kepler problem,
harmonic and anharmonic oscillators.
