  Excited states of Bose--Einstein condensates are considered in the
semi-classical (Thomas-Fermi) limit of the Gross--Pitaevskii equation with
repulsive inter-atomic interactions and a harmonic potential. The relative
dynamics of dark solitons (density dips on the localized condensate) with
respect to the harmonic potential and to each other is approximated using the
averaged Lagrangian method. This permits a complete characterization of the
equilibrium positions of the dark solitons as a function of the chemical
potential parameter. It also yields an analytical handle on the oscillation
frequencies of dark solitons around such equilibria. The asymptotic predictions
are generalized for an arbitrary number of dark solitons and are corroborated
by numerical computations for 2- and 3-soliton configurations.
