  In this paper we use microscopic arguments to derive a nonlinear
Schr\"{o}dinger equation for trapped Bose-condensed gases. This is made
possible by considering the equations of motion of various anomalous averages.
The resulting equation explicitly includes the effect of repeated binary
interactions (in particular ladders) between the atoms. Moreover, under the
conditions that dressing of the intermediate states of a collision can be
ignored, this equation is shown to reduce to the conventional Gross-Pitaevskii
equation in the pseudopotential limit. Extending the treatment, we show first
how the occupation of excited (bare particle) states affects the collisions,
and thus obtain the many-body T-matrix approximation in a trap. In addition, we
discuss how the bare particle many-body T-matrix gets dressed by mean fields
due to condensed and excited atoms. We conclude that the most commonly used
version of the Gross-Pitaevskii equation can only be put on a microscopic basis
for a restrictive range of conditions. For partial condensation, we need to
take account of interactions between condensed and excited atoms, which, in a
consistent formulation, should also be expressed in terms of the many-body
T-matrix. This can be achieved by considering fluctuations around the
condensate mean field beyond those included in the conventional finite
temperature mean field, i.e. Hartree-Fock-Bogoliubov (HFB), theory.
