  Motivated by the fact that two-component confined fermionic gases in
Bardeen-Cooper-Schrieffer-Bose-Einstein condensate (BCS-BEC) crossover can be
described through an hydrodynamical approach, we study these systems - both in
the cigar-shaped configuration and in the disk-shaped one - by using a
polytropic Lagrangian density. We start from the Popov Lagrangian density and
obtain, after a dimensional reduction process, the equations that control the
dynamics of such systems. By solving these equations we study the sound
velocity as a function of the density by analyzing how the dimensionality
affects this velocity
