  By using a variational approach in combination with the adiabatic
approximation we derive a new effective 1D equation of motion for the axial
dynamics of elongated condensates. For condensates with vorticity $|q|=0$ or 1,
this equation coincides with our previous proposal [A. Mu\~noz Mateo and V.
Delgado, Phys. Rev. A \textbf{77}, 013617 (2008)]. We also rederive the
nonpolynomial Schr\"{o}dinger equation (NPSE) in terms of the adiabatic
approximation. This provides a unified treatment for obtaining the different
effective equations and allows appreciating clearly the differences and
similarities between the various proposals. We also obtain an expression for
the axial healing length of cigar-shaped condensates and show that, in the
local density approximation and in units of the axial oscillator length, it
coincides with the inverse of the condensate axial half-length. From this
result it immediately follows the necessary condition for the validity of the
local density approximation. Finally, we obtain analytical formulas that give
the frequency of the axial breathing mode with accuracy better than 1%. These
formulas can be relevant from an experimental point of view since they can be
expressed in terms only of the axial half-length and remain valid in the
crossover between the Thomas-Fermi and the quasi-1D mean-field regimes. We have
corroborated the validity of our results by numerically solving the full 3D
Gross-Pitaevskii equation.
