  In recent years, a number of dynamical density functional theories (DDFTs)
have been developed for describing the dynamics of the one-body density of both
colloidal and atomic fluids. In the colloidal case, the particles are assumed
to have stochastic equations of motion and theories exist for both the case
when the particle motion is over-damped and also in the regime where inertial
effects are relevant. In this paper we extend the theory and explore the
connections between the microscopic DDFT and the equations of motion from
continuum fluid mechanics. In particular, starting from the Kramers equation
which governs the dynamics of the phase space probability distribution function
for the system, we show that one may obtain an approximate DDFT that is a
generalisation of the Euler equation. This DDFT is capable of describing the
dynamics of the fluid density profile down to the scale of the individual
particles. As with previous DDFTs, the dynamical equations require as input the
Helmholtz free energy functional from equilibrium density functional theory
(DFT). For an equilibrium system, the theory predicts the same fluid one-body
density profile as one would obtain from DFT. Making further approximations, we
show that the theory may be used to obtain the mode coupling theory that is
widely used for describing the transition from a liquid to a glassy state.
