  We investigate the interactions of collective excitations with vortices in
superfluid systems, including $~^4$He and superconductors. The dynamical
equations are obtained by the aid of the many-body wavefunction and the
density-density correlation function. The scattering cross-section of
collective excitations with a vortex is calculated in the Born approximation
(valid at long wavelengths), and is expressed in terms of the collective
excitation spectrum $ \omega (q) $ by the simple and general formula $ \sigma =
\frac{\pi}{2} \frac{q}{[\omega'(q)]^2} \cot^2\frac{\theta}{2} $, where $ q $ is
the wave number of the excitation, and $\theta $ is the scattering angle. At
short wave lengths, the classical equations of motion are derived.
