  We investigate the dynamics of a dark-bright soliton in a harmonic potential
using a mean-field approach via coupled nonlinear Schr\"odinger equations
appropriate to multicomponent Bose-Einstein condensates. We use a modified
perturbed dynamical variational Lagrangian approximation, where the
perturbation is due to the trap, taken as a Thomas-Fermi profile. The
wavefunction ansatz is taken as the correct hyperbolic tangent and secant
solutions in the scalar case for the dark and bright components of the soliton,
respectively. We also solve the problem numerically with psuedo-spectral
Runge-Kutta methods. We find, analytically and numerically, for weak trapping
the internal modes are nearly independent of center of mass motion of the
dark-bright soliton. In contrast, in tighter traps the internal modes couple
strongly to the center of mass motion, showing that for dark-bright solitons in
a harmonic potential the center of mass and relative degrees of freedom are not
independent. This result is robust against noise in the initial condition and
should, therefore, be experimentally observable.
