Linear stability of homogeneous and quasi-homogeneous N-body problem by symmetry groups

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.