  We write equations of motion for density variables that are equivalent to
Newtons equations. We then propose a set of trial equations parameterised by
two unknown functions to describe the exact equations. These are chosen to best
fit the exact Newtonian equations. Following established ideas, we choose to
separate these trial functions into a set representing integrable motions of
density waves, and a set containing all effects of non-integrability. It
transpires that the static structure factor is fixed by this minimum condition
to be the solution of the Yvon-Born-Green (YBG) equation. The residual
interactions between density waves are explicitly isolated in their Newtonian
representation and expanded by choosing the dominant objects in the phase space
of the system, that can be represented by a dissipative term with memory and a
random noise. This provides a mapping between deterministic and stochastic
dynamics. Imposing the Fluctuation-Dissipation Theorem (FDT) allows us to
calculate the memory kernel. We write exactly the expression for it, following
two different routes, i.e. using explicitly Newtons equations, or instead,
their implicit form, that must be projected onto density pairs, as in the
development of the well-established Mode Coupling Theory (MCT). We compare
these two ways of proceeding, showing the necessity to enforce a new equation
of constraint for the two schemes to be consistent. Thus, while in the first
`Newtonian' representation a simple gaussian approximation for the random
process leads easily to the Mean Spherical Approximation (MSA) for the statics
and to MCT for the dynamics of the system, in the second case higher levels of
approximation are required to have a fully consistent theory.
