标题： 基于晶体点群的三体问题中八字编队的分岔分析 显示英文标题

标题： Bifurcation analysis of figure-eight choreography in the three-body problem based on crystallographic point groups

摘要： 基于对称群，通过作用量的变分原理分析了figure-eight舞蹈的分岔。 不可约表示决定了Lyapunov-Schmidt约化作用量的对称性和维数，这在分岔级联中产生了四种类型的分岔。 类型1分岔由平凡表示表示，分岔出两个解。 类型2由非平凡的一维表示表示，分岔出两个全等的解。 类型3和4由二维不可约表示表示，分别分岔出三组和六组全等的解。 我们分析了之前发表的数值分岔解和四个新的解：类型2的非对称舞蹈解，类型2的非平面解，类型3的$y$轴对称解，以及类型4的非对称解。 显示英文摘要

摘要： The bifurcation of figure-eight choreography is analyzed by its symmetry group based on the variational principle of the action. The irreducible representations determine the symmetry and the dimension of the Lyapunov-Schmidt reduced action, which yields four types of bifurcations in the sequence of the bifurcation cascade. Type 1 bifurcation, represented by trivial representation, bifurcates two solutions. Type 2, by non-trivial one-dimensional representation, bifurcates two congruent solutions. Type 3 and 4, by two-dimensional irreducible representations, bifurcate two sets of three and six congruent solutions, respectively. We analyze numerical bifurcation solutions previously published and four new ones: non-symmetric choreographic solution of type 2, non-planar solution of type 2, $y$-axis symmetric solution of type 3, and non-symmetric solution of type 4.