  Motivated by Xia-Zhou's recent work on applying symmetry groups to the N-body
problem, we will study relative equilibria of the equilateral triangle and the
square configurations under $\alpha$-homogeneous and quasi-homogeneous
potentials with this method.
  After linearizing the corresponding second order equations, with appropriate
coordinate transformations, we study the linear stability of the relative
equilibria by decomposing each $2n\times 2n$ matrix into a series of $2\times
2$ matrices.

Classical Analysis and ODEs