论文标题
分叉的作用和群体理论的变分原理和群体理论
Variational principle of action and group theory for bifurcation of figure-eight solutions
论文作者
论文摘要
图八溶液是平面在同质或不均匀电位下平面相等质量三体问题的解决方案。众所周知，在转换组$ d_6 $：常规六角形的二面体集团下，它们是不变的。数值研究表明，每个图八溶液都有一些分叉点。相对于分叉溶液的对称性，已知六个分叉模式。 在本文中，我们将展示以下内容。动作和群体理论的变分原理表明，每个人物八解决方案的分叉都取决于$ d_6 $的不可约表示。每个不可约表示对每个分叉都有一对一的对应关系。这解释了数值观察到的六个分叉模式。通常，在拉格朗日力学中，周期性解决方案的分叉是由使该溶液不变的转换组的不可还原表示确定的。
Figure-eight solutions are solutions to planar equal mass three-body problem under homogeneous or inhomogeneous potentials. They are known to be invariant under the transformation group $D_6$: the dihedral group of regular hexagons. Numerical investigation shows that each figure-eight solution has some bifurcation points. Six bifurcation patterns are known with respect to the symmetry of the bifurcated solution. In this paper we will show the followings. The variational principle of action and group theory show that the bifurcations of every figure-eight solution are determined by the irreducible representations of $D_6$. Each irreducible representation has one to one correspondence to each bifurcation. This explains numerically observed six bifurcation patterns. In general, in Lagrangian mechanics, bifurcations of a periodic solution is determined by irreducible representations of the transformation group that leaves this solution invariant.
