Stability of small planar dust cluster in the presence of an external magnetic field

Oscillations and stability of planar symmetric dust clusters shaped as a regular polygon and a regular polygon with a particle at the center in the presence of an external uniform magnetic field are analyzed. The structure stability against small perturbations is considered. To this end, the equation of motion was linearized with respect to small coordinate variations. As a result, a dispersion relation (quartic algebraic equation) was derived and numerically studied for particular types of interactions between particles. A comparison of stability regions for a quadrangle and a triangle with a particle at the center, as well as a pentagon and a quadrangle with a particle at the center shows that there are ranges of parameters, where configurations without particle at the center are stable and configurations with a particle at the center are unstable, and vice versa.