  We present a method for approximating the solution of the three-dimensional,
time-dependent Gross-Pitaevskii equation (GPE) for Bose-Einstein condensate
systems where the confinement in one dimension is much tighter than in the
other two. This method employs a hybrid Lagrangian variational technique whose
trial wave function is the product of a completely unspecified function of the
coordinates in the plane of weak confinement and a gaussian in the strongly
confined direction having a time-dependent width and quadratic phase. The
hybrid Lagrangian variational method produces equations of motion that consist
of (1) a two-dimensional, effective GPE whose nonlinear coefficient contains
the width of the gaussian and (2) an equation of motion for the width that
depends on the integral of the fourth power of the solution of the 2D effective
GPE. We apply this method to the dynamics of Bose-Einstein condensates confined
in ring-shaped potentials and compare the approximate solution to the numerical
solution of the full 3D GPE.
