三体问题

In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.[1] The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

在物理学和经典力学领域中，三体问题是根据牛顿运动定律和万有引力定律计算三点质量的初始位置和速度（或动量）并求出它们随后的运动的问题。三体是n体问题中的一个特例。与两体不同的是，三体问题不存在通用的封闭形式的解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以需要数值方法。

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, and the Sun.[2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

历史上看，第一个被研究的特定三体问题是月球、地球和太阳构成的“三体”。扩展后三体问题的现代意义是经典力学或量子力学中模拟三个粒子运动的任何问题。

数学描述

三体的数学表达式可以用三个质量为[math]\displaystyle{ m_i }[/math]相互作用的物体的矢量位置[math]\displaystyle{ \mathbf{r_i} = (x_i, y_i, z_i) }[/math]的牛顿运动方程来表示：

[math]\displaystyle{ \begin{align} \ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\ \ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\ \ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}. \end{align} }[/math]

其中[math]\displaystyle{ G }[/math]为万有引力常数。这是一组9个二阶微分方程。这个问题也可以用哈密顿形式等价表示，此时可以用一组18个一阶微分方程来描述，这些方程分别对应于位置[math]\displaystyle{ \mathbf{r_i} }[/math]和动量[math]\displaystyle{ \mathbf{p_i} }[/math]的一个分量：

[math]\displaystyle{ \frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}}, }[/math]

其中[math]\displaystyle{ \mathcal{H} }[/math]是 Hamiltonian 函数：

[math]\displaystyle{ \mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}. }[/math]

这种情况下，[math]\displaystyle{ \mathcal{H} }[/math]仅仅是系统的总能量，重力加上动能。

受限制的三题问题

In the restricted three-body problem,[3] a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the force that the planetoid exerts on the two massive bodies may be neglected, and the system can be analysed and can therefore be described in terms of a two-body motion. Usually this two-body motion is taken to consist of circular orbits around the center of mass, and the planetoid is assumed to move in the plane defined by the circular orbits.

在受限制的三体问题中，一个质量可忽略不计的天体(“小行星”)在两个质量巨大的天体的影响下运动。由于质量可忽略不计，小行星对这两个质量巨大的物体所施加的力可忽略不计，因此可以可以用两个物体的运动来描述，对该系统进行分析。通常这种两体运动被认为是由围绕质心的圆形轨道组成的，并且假定小行星在圆形轨道所定义的平面内运动。

The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.

有限的三体问题比完整的问题更容易从理论上分析。它也具有实际意义，因为它准确地描述了许多现实世界的问题，其中最重要的例子是地球-月亮-太阳系，这也是在三体问题的发展历史中有重要地位的一个典型。

Mathematically, the problem is stated as follows. Let {\displaystyle m_{1,2}} {\displaystyle m_{1,2}} be the masses of the two massive bodies, with (planar) coordinates {\displaystyle (x_{1},y_{1})} (x_{1},y_{1}) and {\displaystyle (x_{2},y_{2})} (x_{2},y_{2}), and let {\displaystyle (x,y)} (x,y) be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to {\displaystyle 1} 1. Then, the motion of the planetoid is given by

在数学的表述上，设[math]\displaystyle{ m_{1,2} }[/math]为两个小行星的质量，二维平面坐标[math]\displaystyle{ (x_1, y_1) }[/math]和[math]\displaystyle{ (x_2, y_2) }[/math]分别为小行星的坐标。简单起见，选择的单位应该要确保两小行星的距离和重力常数都等于1。则小行星的运动可以用公式描述：

[math]\displaystyle{ \begin{align} \frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3} \\ \frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3}, \end{align} }[/math]

其中[math]\displaystyle{ r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2} }[/math]，在这种形式下，运动方程通过坐标具有明确的时间依赖性[math]\displaystyle{ x_i(t), y_i(t) }[/math]。但可以通过转换为旋转参考系来消除这种时间相关性，从而简化了后续的分析。

求解

There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals.[1] Moreover, the motion of three bodies is generally non-repeating, except in special cases.[5]

由简单的代数表达式和积分给出的三体没有一般的解析解。此外，除特殊情况，三个物体的运动一般是不重复的。

On the other hand, in 1912 the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t1/3 for the 3-body problem.[6] This series converges for all real t, except for initial conditions corresponding to zero angular momentum. (In practice the latter restriction is insignificant since such initial conditions are rare, having Lebesgue measure zero.)

另一方面，1912年芬兰数学家**Karl Fritiof Sundman**证明了三体问题存在一个 t^1/3次方的级数解。除了对应于零角动量的初始条件外，这个级数对所有实数t都收敛。

An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Therefore, it is necessary to study the possible singularities of the 3-body problems. As it will be briefly discussed below, the only singularities in the 3-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

证明这个结果的一个重要问题是，该序列的收敛半径是由到最近奇点的距离决定的。因此，有必要研究三体问题的可能奇点。三体问题中唯一的奇点是双碰撞(两个粒子在瞬间的碰撞)和三重碰撞(三个粒子在瞬间的碰撞)。

Collisions, whether binary or triple (in fact, any number), are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. However, there is no criterion known to be put on the initial state in order to avoid collisions for the corresponding solution. So Sundman's strategy consisted of the following steps:

无论是二元的还是三元的(事实上是任何数目) 碰撞都不太可能发生，因为已经证明它们对应于测度为零的一组初始条件。然而，没有已知的标准被放在初始状态，以避免相应的解决方案碰撞。因此，**Sundman**的求解方法包括以下步骤:

1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization. 2. Proving that triple collisions only occur when the angular momentum L vanishes. By restricting the initial data to L ≠ 0, he removed all real singularities from the transformed equations for the 3-body problem. 3. Showing that if L ≠ 0, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by using Cauchy's existence theorem for differential equations, that there are no complex singularities in a strip (depending on the value of L) in the complex plane centered around the real axis (shades of Kovalevskaya). 4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t1/3 (the new variable after the regularization) and if |ln s| ≤ β,[clarification needed] then this map is given by

1. 使用适当的变量更改来继续分析二进制冲突之外的解决方案，这个过程称为正则化。

2. 证明只有在角动量L消失时才会发生三元碰撞。通过将初始数据限制为L ≠ 0，从三体问题的变换方程中删除了所有实际奇点。

3. 证明了如果L≠0，则不仅不存在三元碰撞，而且系统严格有界远离三元碰撞。这意味着，通过使用柯西微分方程的存在性定理，在以实际轴为中心的复平面（Kovalevskaya的阴影）中，条带（取决于L的值）中不存在复奇点。

4. 找到一个保角变换，把这个条带映射到单位圆盘。例如，如果s=t^1/3（正则化后的新变量），并且如果模板:Abs ≤ β（需要证明），则可由下式给出：

[math]\displaystyle{ \sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}. }[/math]

This finishes the proof of Sundman's theorem. 上述即为完整的Sundman定力的证明。

Unfortunately, the corresponding series converges very slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 108000000 terms.[7]

但对应的序列收敛得非常慢。也就是说，为了没有太多实际用途的解法，需要很多的附加项的相关量的求解才能得到有意义的精度。如1930年****，David Beloriszky计算出，如果将Sundman的级数用于天文观测，则计算将至少涉及10^模板:Val项。

特殊的求解方法

In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. See Euler's three-body problem.

1767年，**Leonhard Euler**提出了三个周期解系列，其中三个质量在每个瞬间共线。

1772年，拉格朗日（Lagrange）找到了一系列解，其中三个质量在每个瞬间形成一个等边三角形。这些解决方案与Euler的共线解一起构成了三体问题的中心配置。这些解决方案对于任何质量比均有效，并且质量沿开普勒椭圆形运动。这四个族是唯一有明确解析公式的已知解决方案。在圆形受限三体问题的特殊情况下，这些解决方案在与原边一起旋转的框架中观察时，变为称为L₁, L₂, L₃, L₄和L₅，并且叫做拉格朗日点，其中L₃, L₄是拉格朗日的对称解的实例。

在1892年至1899年的工作中，**Henri Poincaré**建立了无穷有限三体问题的周期解，以及将这些解法继续推广到一般三体问题的技巧。

1893年，迈塞尔提出了现在所说的毕达哥拉斯三体问题：将比例为3：4：5的三个质量置于3：4：5直角三角形的顶点处。Burrau在1913年进一步研究了这个问题。1967年，Victor Szebehely和C. Frederick Peters利用数值积分理论建立了这个问题的最终逃逸模型，同时找到了附近的周期解。

20世纪70年代，**Michel Hénon**和 Roger A. Broucke各自找到了一套解决方案，这些解决方案构成了同一系列解决方案的一部分: Broucke–Henon–Hadjidemetriou族。在这个家族中，这三个物体都具有相同的质量，可以表现出逆行和直行两种形式。在、Broucke的一些解中，两个物体遵循同样的路径。

1993年，圣塔菲研究所的物理学家Cris Moore提出了一种零角动量解，该解适用于三个相等质量围绕一个八字形运动。这种方法在2000年由数学家Alain Chenciner和Richard Montgomery证明。在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但有人认为不太可能发生这种情况，因为稳定​​性的范围小。例如，1993年，圣达菲研究所的物理学家克里斯·摩尔（Cris Moore）在数字上发现了一个零角动量解，该解的三个相等质量围绕一个八字形运动。[12]它的正式存在后来在2000年由数学家Alain Chenciner和Richard Montgomery 证明。[13] [14]在数值上证明了该解对于质量和轨道参数的小扰动是稳定的，这增加了在物理宇宙中可以观察到这种轨道的可能性。但是，由于稳定​​性的范围小，因此不太可能发生这种情况。例如，二元-二元散射事件导标号-8轨道的概率估计为1%的一小部分。

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.

2013年，贝尔格莱德物理研究所的物理学家Milovan uvakov 和 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新解。

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.

2015年，物理学家 Ana Hudomal 发现了14种等质量零角动量三体问题的新解。

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.[17] This was followed in 2018 by an additional 1223 new solutions for a zero-momentum system of unequal masses.[18]

2017年，研究人员Xiaoming Li 和Shijun Liao发现了669个等质量零角动量三体问题的新周期轨道。2018年，不等质量的零动量系统又增加了1223个新解。

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three body problem.[19] The free fall formulation of the three body problem starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forwards and backwards along an open "track".

2018年，Li和Liao提出了234个解决不等质量“自由落体”三体问题的方案。三体问题的自由落体公式从所有三个静止的物体开始。正因为如此，质量在一个自由落体配置不在一个闭合的“循环”轨道上运行，而是沿着一个开放的“轨道”向前和向后运行。

数值方法

Using a computer, the problem may be solved to arbitrarily high precision using numerical integration although high precision requires a large amount of CPU time. In 2019, Breen et al. announced a fast neural network solver, trained using a numerical integrator.[20]

使用计算机，尽管高精度需要大量的CPU时间，但是可以使用数值积分将问题解决为任意高精度。在2019年，Breen等人。提出了一种快速的神经网络求解器，使用数字积分器对其进行训练。

历史

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia (Philosophiæ Naturalis Principia Mathematica). In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of the Earth and the Sun.

传统意义上的三个物体的引力问题可以追溯到1687年，当时牛顿发表了他的《自然哲学的数学原理》。在《原理》第一卷的第66号提案及其22个推论中，牛顿首次定义和研究了三个受相互扰动的重力吸引力影响的巨大物体的运动问题。在第三册的第25至35条命题中，牛顿也采取了第一步，将他的66号提案的结果应用到月球理论中，即月球在地球和太阳的引力影响下的运动。

The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei; in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the determination of longitude at sea, solved in practice by John Harrison's invention of the marine chronometer. However the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth.

Amerigo Vespucci和随后的Galileo Galilei提出了三体问题; 1499年，Vespucci利用对月球位置的了解来确定自己在巴西的位置。因为这种方法适用于导航，特别是海上精度，1720年代该方法变得非常技术实用。这种解法在实际中是John Harrison发明的海洋计时器的解决方案。但是，由于太阳和行星对月球绕地球运动的干扰作用，月球理论的准确性很低。

Jean le Rond d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality; they submitted their competing first analyses to the Académie Royale des Sciences in 1747.[21] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: Problème des trois Corps) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.