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arXiv · 2404.08790

Central Configurations with Dihedral Symmetry

Tingjie Zhou·Zhihong Xia
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Abstract

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.

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Tingjie Zhou, Zhihong Xia. 2024-04-12. Central Configurations with Dihedral Symmetry. https://arxiv.org/abs/2404.08790

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