  We parameterize the phase space density by time dependent diffeomorphic,
Poisson preserving transformations on phase space acting on a reference density
solution. We can look at these as transformations which fix time on the
extended space of phase space and time. In this formulation the Vlasov equation
is replaced by a constraint equation for the above maps. The new equations are
formulated in terms of hamiltonian generators of one parameter families of
diffeomorphic, Poisson preserving maps e.g. generators with respect to time or
a perturbation parameter. We also show that it is possible to parameterize the
space of solutions of the Vlasov equation by composition of maps subject to
certain compatibility conditions on the generators. By using this composition
principle we show how to formulate new equations for a hybrid fluid kinetic
theory. This is done by observing that a certain subgroup of the group of phase
space maps with generators which are linear in momentum correspond to the group
of diffeomorphic maps parameterizing the continuity equation in fluid theory.
