The linear stability of the principle relative equilibria in the Coulomb $(n+1)$ body problem

The linear stability of the principle relative equilibria of the Coulomb $(n+1)$-body problem is studied. The $n$ particles are the electrons having a charge of -1 and the $(n+1)^{th}$ particle is the nucleus having a positive charge $Z$ equal to the atomic number. The mass of the nucleus is over three orders of magnitude greater than that of the electrons. Treating the $n$ electrons as identical particles allows the introduction of symmetry variables. This block diagonalizes the Jacobian matrix and consequently factors the characteristic polynomial.