  The dynamics of Bose-Einstein condensation in a three-dimensional harmonic
trap is studied explicitly including the Bogoliubov approximation for
temperatures below the critical one. To model the evolution towards equilibrium
at each cooling step, we derive quantum kinetic equations that describe the
dynamics of the gas for temperatures above and below the transition
temperature. These equations, valid in the Born and Markov approximations,
consider the essential role of the chemical potential as the main parameter
that signals the transition. The kinetic equation that describes the growth of
the condensate below the transition temperature is derived within the
Bogoliubov approximation. To illustrate our results we propose an energetic
cooling protocol and simulate the whole sequence of the formation of a
condensate.
