  We consider a cylindrically symmetric trap containing a small Bose-Einstein
condensate with a singly quantized vortex on the axis of symmetry. A
time-dependent variational Lagrangian analysis yields the small-amplitude
dynamics of the vortex and the condensate, directly determining the equations
of motion of the coupled normal modes. As found previously from the Bogoliubov
equations, there are two rigid dipole modes and one anomalous mode with a
negative frequency when seen in the laboratory frame.
