  This paper concerns an investigation of the spin wave excitations in magnetic
nanoparticles. We provide a detailed derivation of the theoretical method for
the determination of the normal modes of confined magnetic systems based on a
discrete lattice of magnetic moments. The method is based on the damping-free
Landau-Lifshitz equation and general enough to be utilized for the magnetic
system of any dimensionality, magnetic structure, shape, and size. As an
example we explore the influence of the competition between exchange and
dipolar interactions on the spectrum of normal modes as well as on the
stability of the vortex state in two-dimensional nanorings. We show the
lowest-frequency mode to be indicative of the dipolar-to-exchange iterations
ratio. We also study behavior of the fundamental mode and present the influence
of both, the discreteness of the lattice and the dipolar-to-exchange iterations
ratio, on its hybridization with azimuthal modes. We complete the paper with a
selective review of the spin wave excitations in circular dots to compare with
the results obtained for the rings.
