  An ideal compressible fluid is considered, with an equilibrium density being
a given function of coordinates due to presence of some static external forces.
The slow flows in such system, which do not disturb the density, are
investigated with the help of the Hamiltonian formalism. The equations of
motion of the system are derived for an arbitrary given topology of the
vorticity field. The general form of the Lagrangian for frozen-in vortex lines
is established. The local induction approximation for motion of slender vortex
filaments in several inhomogeneous physical models is studied.
