2024/04/12 by Zhou, Tingjie, Xia, Zhihong
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2404.08790
As an application of the representation theory for the dihedral groups, we study the symmetric central configurations in the n-body problem where n equal masses are placed at the vertices of a regular n-gon. Since the Hessian matrices at these configurations are typically very large, particularly when n is large, computations of their eigenvalues present a challenging problem. However, by decomposing the action of the dihedral groups into irreducible representations, we show that the Hessians can be simplified to a block-diagonal matrix with small blocks, of the sizes at most 2*2. This is due to the fact that the action of a dihedral group can be represented as a block-diagonal matrix with small irreducible blocks. In the end, the eigenvalues can be explicitly obtained by simply computing eigenvalues of these small block matrices.