经典 $n$ 体系统在几何变量和体积变量中的描述 I. 三体情形
Classical $n$-body system in geometrical and volume variables. I. Three-body case
AI总结：
本文在零总角动量下，引入相对距离平方和几何变量描述经典三体系统，证明动能不依赖自由度，并给出牛顿引力、双纽线编队和谐振子三个示例。
摘要：
我们考虑具有 $d$ 个自由度（$d>1$）且总角动量为零的经典三体系统。研究仅限于势函数 $V$ 仅依赖于物体之间的相对（相互）距离 $r_{ij}=\mid {\bf r}_i - {\bf r}_j\mid$ 的情形。遵循 J. L. Lagrange 的提议，在质心系中我们引入相对距离（辅以角度）作为广义坐标，并证明动能不依赖于 $d$，这证实了 Murnaghan（1936）在 $d=2$ 和 van Kampen-Wintner（1937）在 $d=3$ 的结果，在 $d=3$ 时它对应于一个三维刚体。认识到 $\mathbb{Z}_2$ 对称性（$r_{ij} \rightarrow -r_{ij}$），我们引入新变量 $ρ_{ij}=r_{ij}^2$，这使得惯性张量在二体碰撞时非奇异。在这些变量下，动能是 $ρ$ 相空间中的多项式函数。三个物体的位置构成一个（相互作用的）三角形，且动能关于物体位置和质量的交换（以及三角形边和质量的交换）是 $\mathcal{S}_3$ 置换不变的。对于等质量情形，我们使用 $\mathbb{Z}_2^{\otimes3} \oplus \mathcal{S}_3$ 的最低阶对称多项式不变量来定义新的广义坐标，称为几何变量。其中两个最低阶的（三角形边长的平方和与面积的平方）称为体积变量。我们详细研究三个例子：（I）$d=3$ 时的三体牛顿引力，（II）$d=2$ 时 Fujiwara 等人提出的代数双纽线上的三体编队运动，其中问题在几何变量下变为一维，以及（III）（非）谐振子。
英文摘要：
We consider the classical 3-body system with $d$ degrees of freedom $(d>1)$ at zero total angular momentum. The study is restricted to potentials $V$ that depend solely on relative (mutual) distances $r_{ij}=\mid {\bf r}_i - {\bf r}_j\mid$ between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on $d$, confirming results by Murnaghan (1936) at $d=2$ and van Kampen-Wintner (1937) at $d=3$, where it corresponds to a 3D solid body. Realizing $\mathbb{Z}_2$-symmetry $(r_{ij} \rightarrow -r_{ij})$ we introduce new variables $ρ_{ij}=r_{ij}^2$, which allows us to make the tensor of inertia non-singular for binary collisions. In these variables the kinetic energy is a polynomial function in the $ρ$-phase space. The 3 body positions form a triangle (of interaction) and the kinetic energy is $\mathcal{S}_3$-permutationally invariant wrt interchange of body positions and masses (as well as wrt interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of $\mathbb{Z}_2^{\otimes3} \oplus \mathcal{S}_3$ to define new generalized coordinates, they are called the {\it geometrical variables}. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called {\it volume variables}. We study three examples in some detail: (I) 3-body Newton gravity in $d=3$, (II) 3-body choreography in $d=2$ on the algebraic lemniscate by Fujiwara et al where the problem becomes one-dimensional in the geometrical variables, and (III) the (an)harmonic oscillator.
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