  We derive the coupled equations of motion for the condensate (superfluid) and
non-condensate (normal fluid) degrees of freedom in a trapped Bose gas at
finite temperatures. Our results are based on the Hartree-Fock-Popov
approximation for the time-dependent condensate wavefunction and an assumption
of local equilibrium for the non-condensate atoms. In the case of a uniform
weakly-interacting gas, our formalism gives a microscopic derivation of the
well-known two-fluid equations of Landau. The collective modes in a
parabolically trapped Bose gas include the analogue of the out-of-phase second
sound mode in uniform systems.
