  We discuss the collective excitations in a spatially inhomogeneous
(cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature
($T \ll T_{\lambda}$). The main result is the dispersion relation for a (first)
sound wave that is obtained by describing the perturbation as a Lagrangian
coordinate. The dispersion curve is in good agreement with the Bogoliubov
phonon spectrum $\omega=c k$, where $k=k_{\theta}=m/r$, the wave number and
$c=[4 \pi a \rho \hbar^{2}]^{1/2}/M$, the speed of first sound. Based on Bohm's
quantum theory, a spontaneously broken gauge symmetry in a quantum fluid is
discussed in terms of the quantum fluctuation-dissipation, from which it is
shown that the symmetry breaking takes place at the free surface of BEC in an
external field.
