APPLYING THE SYMMETRY GROUPS TO STUDY THE N-BODY PROBLEM

对称群在牛顿n体问题中的应用

周婷洁

11930530

硕士

数学

夏志宏

2021-05-20

2021-05-20

南方科技大学

深圳

We introduce an algebra method to study the Newtonian 𝑛-body problem. The mainidea is considering the symmetry of the central configurations and using the representationtheory of groups to simplify the calculation of the eigenvalues. When the 𝑛 particleshave equal masses, it has the regular 𝑛-gon as a solution which is a central configuration.The dihedral group with degree 𝑛 is the symmetry group for the regular 𝑛-gon. Everyelement in the 𝑛 dihedral group can be a linear transformation acting on the regular 𝑛-gon. As matrices, it is a group representation for the dihedral group. We have known allthe irreducible group representations for the dihedral group. Therefore, we can decomposethe group representation into the direct sum of irreducible group representations.Due to the symmetry of the central configuration, the Hessian for some smooth functionsis invariant under the 𝑛 dihedral group. Choosing eigenvectors as new coordinates, thegroup representation will be in a special form which is equivalent to the direct sum ofirreducible group representations. With the character table of the symmetry group, it iseasy to get linear equations for eigenvalues. By solving equations, we can get informationfor eigenvalues which is hard to obtain traditionally.For the 4-body problem with particles have the same masses, squares are a centralconfiguration. We can obtain eigenvalues for the Hessian by simple calculation with tracein the group representation. For the 4-body problem with three particles have the samemasses and another particle has mass 𝑚, we study the Hessian at the center + equilateraltriangle configuration. This technique can give the relationship between eigenvalues andthe parameter 𝑚. It proves the existence of a degenerate central configuration for the4 body problem. For the Hamiltonian, there are eigenvalues that have real part. Thenthe center + equilateral triangle configuration configuration for the 4-body problem isunstable. Moreover, the number of the eigenvalues in the ellipse direction changes from6 to 10 when 𝑚 is large enough.

我们借助代数中群表示等经典理论有效地研究了牛顿n-体问题。主要的思想是利用中心构型的对称性和群表示简化特征根的计算。当n-体问题中的天体具有相同的质量时，它存在正n 边形作为特解。同时，正n边形为n-体问题的一类中心构型。它具有度为𝑛 的二面体群作为对称群。二面体群的每个元素可以作用在正𝑛边形上，对其进行旋转、反射等变换。在线性空间中，我们用矩阵形式描述这些变换，这构成了二面体群的一个群表示。已知二面体群的所有不可约群表示，因而可以把此群表示分解成不可约群表示的直和。对于可与这些线性变换矩阵交换的矩阵，我们说矩阵在群作用下保持不变。选取矩阵的特征向量作为新坐标，可以推知矩阵的特殊结构。最后结合有限群的特征表，我们可以得到关于特征值的线性方程组，并解得相应的特征值。等质量的4-体问题有一类正方形中心构型。此中心构型对应的海瑟矩阵的特征根可以通过群表示进行简单的迹运算得到。对于4-体问题，我们研究有三个等质量天体和一个带有质量参数𝑚 的天体的情况，其具有一类中心+ 等边三角形构型。本文研究了在中心+ 等边三角形类构型处的海瑟矩阵，得到其特征值和参数𝑚 的关系，证明了退化相对平衡解在4-体问题中的存在性。此外，我们发现4-体问题对应的哈密顿系统在中心+ 等边三角形构型出存在具有实部的特征值，因此中心+ 等边三角形构型的中心构型是不稳定的。并且当中心天体质量m 充分大时，椭圆方向的特征值由6 个变成了10 个。

the Newtonian n-body problem symmetry the representation theory eigenvalues

牛顿n-体问题 对称性 群表示 特征值

英语

独立培养

南方科技大学

Zhou TJ. APPLYING THE SYMMETRY GROUPS TO STUDY THE N-BODY PROBLEM[D]. 深圳. 南方科技大学,2021.