  We present a numerical study of the effect of the repulsive logarithmic
inter-particle interaction on the ground state configuration and the frequency
spectrum of a confined classical two-dimensional cluster containing a finite
number of particles. In the case of a hard wall confinement all particles form
one ring situated at the boundary of the potential. For a general r^n
confinement potential, also inner rings can form and we find that all
frequencies lie below the frequency of a particular mode, namely the
breathing-like mode. An interesting situation arises for the parabolic confined
system(i.e. n=2). In this case the frequency of the breathing mode is
independent of the number of particles leading to an upper bound for all
frequencies. All results can be understood from Earnshaw's theorem in two
dimensions. In order to check the sensitivity of these results, the spectrum of
vortices in a type II superconductor which, in the limit of large penetration
depths, interact through a logarithmic potential, is investigated.
