  With use of the nonlinear Schr{\"o}dinger (or Gross-Pitaevskii) equation with
strong repulsive cubic nonlinearity, dynamics of multi-component Bose-Einstein
condensates (BECs) with a harmonic trap in 2 dimensions is investigated beyond
the Thomas-Fermi regime. In the case when each component has a single vortex,
we obtain an effective nonlinear dynamics for vortex cores (particles). The
particles here acquire the inertia, in marked contrast to the standard theory
of point vortices widely known in the usual hydrodynamics. The effective
dynamics is equivalent to that of charged particles under a strong spring force
and in the presence of Lorentz force with the uniform magnetic field. The
inter-particle (vortex-vortex) interaction is singularly-repulsive and
short-ranged with its magnitude decreasing with increasing distance of the
center of mass from the trapping center. "Chaos in the three-body problem" in
the three vortices system can be seen, which is not expected in the
corresponding point vortices without inertia in 2 dimensions.
