We calculate the magnon modes in the presence of a vortex on a circular system, combining analytical calculations in the continuum limit with a numerical diagonalization of the discrete system. The magnon modes are expressed by the S matrix for magnon-vortex scattering, as a function of the parameters and the size of the system and for different boundary conditions. Certain quasilocal translational modes are identified with the Frequencies which appear in the trajectory (X) over right arrow(t) of the vortex center in recent molecular dynamics simulations of the full many-spin model. Using these quasilocal modes we calculate the two parameters of a third-order equation of motion for (X) over right arrow(t). This equation was recently derived by a collective variable theory and describes very well the trajectories observed in the simulations. Both parameters, the vortex mass and the factor in front of (X) over right arrow, depend strongly on the boundary conditions. [References: 27]
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