  The variational method of coupled Gaussian functions is applied to
Bose-Einstein condensates with long-range interactions. The time-dependence of
the condensate is described by dynamical equations for the variational
parameters. We present the method and analytically derive the dynamical
equations from the time-dependent Gross-Pitaevskii equation. The stability of
the solutions is investigated using methods of nonlinear dynamics. The concept
presented in this paper will be applied to Bose-Einstein condensates with
monopolar 1/r and dipolar 1/r^3 interaction in the subsequent paper [S. Rau et
al., Phys. Rev. A, submitted], where we will present a wealth of new phenomena
obtained by using the ansatz with coupled Gaussian functions.
