  Starting from an approximate microscopic model of a trapped Bose-condensed
gas at finite temperatures, we derive an equation of motion for the condensate
wavefunction and a quantum kinetic equation for the distribution function for
the excited atoms. The kinetic equation includes collisions between the
condensate and non-condensate atoms ($C_{12}$), in addition to collisions
between the excited atoms as described by the Uehling-Uhlenbeck ($C_{22}$)
collision integral. Assuming that the $C_{22}$ collision rate is sufficiently
rapid to produce a local equilibrium Bose distribution, the kinetic equation
can be used to derive hydrodynamic equations for the non-condensate. These
equations include a description of the equilibration of the local chemical
potentials of the condensate and non-condensate components which gives rise to
a new relaxational mode associated with the exchange of atoms between the two
components. We show how the Landau two-fluid equations emerge in the frequency
domain $\omega \tau_\mu \ll 1$, where $\tau_\mu$ is a characteristic relaxation
time of the equilibration process. This process provides an additional source
of damping of the collective modes (first and second sound in the case of a
uniform system). Our equations are consistent with the generalized Kohn
theorem. Finally, a variational solution of the equations is developed which is
used to determine some of the monopole, dipole and quadrupole normal modes of a
trapped Bose gas in an isotropic trap.
