We consider the classical three-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 rij = |ri −rj| 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 ℤ2-symmetry (rij →−rij), we introduce new variables ρij = rij2, which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the ρ-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is 𝒮3-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of ℤ2⊗3 ⊕𝒮 3 to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in d = 3, (II) three-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.

中文翻译：

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几何和体积变量中的经典 n 体系统：I. 三体情况

我们考虑经典的三体系统d自由程度(d > 1)总角动量为零。该研究仅限于潜力五仅取决于相对（相互）距离r一世j = |r一世 -rj|身体之间。根据 JL Lagrange 的建议，在质心框架中，我们将相对距离（由角度补充）作为广义坐标引入，并表明动能不依赖于d，证实了 Murnaghan (1936) 在d = 2和范坎彭-温特纳 (1937) 在d = 3，它对应于一个 3D 实体。实现ℤ2-对称(r一世j →-r一世j), 我们引入新变量ρ一世j = r一世j2，这使我们能够使二元碰撞的惯性张量非奇异。在这些变量中，动能是一个多项式函数ρ-相空间。三体位置形成（相互作用的）三角形，动能为𝒮3-关于身体位置和质量的交换（以及关于三角形的边缘和质量的交换）的置换不变。对于相等的质量，我们使用最低阶对称多项式不变量ℤ2⊗3 ⊕𝒮 3为了定义新的广义坐标，它们被称为几何变量. 其中两个最低阶（三角形边的平方和和面积的平方）称为体积变量. 结果表明，对于仅取决于几何变量 (i) 和仅取决于质量相关体积变量 (ii) 的势能，可以认为 Hamilton 运动方程相对简单。我们比较详细地研究了三个例子：（一）三体牛顿引力在d = 3, (II) 三体舞蹈d = 2藤原在代数双线上等。，其中问题在几何变量和（III）（an）谐波振荡器中变为一维。