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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.


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.



三体的数学表达式可以用三个质量为[math]\displaystyle{ m_i }[/math]的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个二阶微分方程构成的方程组。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]是哈密顿 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。


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.


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.


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]为两个大质量天体的质量,二维平面坐标[math]\displaystyle{ (x_1, y_1) }[/math][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大质量天体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为:

[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。


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]


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 卡尔·弗里蒂奥夫·桑德曼 Karl Fritiof Sundman 证明了三体问题存在一个 t1/3幂次方的级数解。除了对应于角动量为零的初始条件外,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).


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,以对相应的解避免碰撞。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. 使用适当的变量变化来继续分析二元碰撞之外的解,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数奇点。

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,通过对微分方程使用柯西存在性定理,在以实际轴为中心的复平面(Kovalevskaya的阴影)中,一个条带区域(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=t1/3(正则化后的新变量),并且模板:Absβ(需要证明),则映射可由下式给出:

[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定律的证明。

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]

但不幸运的是,对应的级数收敛得非常慢。也就是说,为了获得一定精度的值需要很多级数项,这样的解法并没有什么实际用途。的确,在1930年,大卫·贝洛里奇 David Beloriszky计算出,如果将Sundman级数用于天文观测,则计算将至少涉及10模板:Val项。


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, 莱昂哈德·欧拉 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。这些解决方案与欧拉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,亨利·波因加 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。布鲁 Burrau在1913年进一步研究了这个问题。1967年,维克多·塞贝赫利 Victor Szebehely和 C、 弗雷德里克·彼得斯 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,米歇尔·赫农 Michel Hénon和 罗杰A.布鲁克 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: 布鲁克-赫农-哈德吉德梅特里奥 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。在布鲁克的一些解中,两个物体遵循同样的路径。

1993年,圣塔菲研究所的物理学家克里斯摩尔 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年由数学家阿兰·契纳 Alain Chenciner理查德·蒙哥马利 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 米洛万·乌瓦科夫 Milovan uvakov 和 维利科·德米特拉·伊诺维 Veljko dmitra inovi 发现了等质量零角动量三体问题的13种新的解族。

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 安娜·胡多马尔 Ana Hudomal 发现了14种等质量零角动量三体问题的新解族。

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 李晓明 Xiaoming Li 和 廖世俊 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,李晓明 Li 和 廖世俊 Liao 提出了234个不等质量“自由落体”三体问题的解。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。


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]

尽管高精度需要大量的CPU时间,但是通过计算机可以使用数值积分可以得到问题的任意高精度解。在2019年,布林 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。


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 艾萨克·牛顿 Isaac Newton 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,牛顿也迈出了第一步,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.

亚美利哥·韦斯普奇 Amerigo Vespucci和随后的 伽利略·伽利雷 Galileo Galilei提出了三体问题; 1499年,韦斯普奇 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,特别是在海上确定经度,1720年代该方法变得非常技术实用。事实上确定经度的问题被 约翰·哈里森 John Harrison发明的航海经线仪所解决。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.

建立了长期竞争关系的 让·勒朗·达朗贝尔 Jean le Rond d'Alembert 和 亚历克西斯·克莱奥特 Alexis Clairaut都试图以某种普遍性来分析该问题。他们于1747年向皇家科学研究院提交了他们的第一批竞争分析。在1740年代的巴黎,“三体问题”(法语:Problèmedes trois Corps)这个名字开始被普遍使用,与他们的研究有关。让·勒朗·达朗贝尔 Jean le Rond d'Alembert于1761年发布的文章表明该名称最早于1747年使用。


The term 'three-body problem' is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies. 三体问题这个术语有时用在更一般的意义上来指涉及三个物体相互作用的任何物理问题。

A quantum mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.[23]


In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force which do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.[24] This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.[24][25]


The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.[26]



The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in the form of a convergent power series, as was proven by Karl F. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;[27] therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see n-body simulation). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the n-body problem usually refers to a galaxy or to a cluster of galaxies; planetary systems, such as stars, planets, and their satellites, can also be treated as n-body systems. Some applications are conveniently treated by perturbation theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.

三体问题是N体问题的一个特例,它描述了n个物体在其中一种物理力(如重力)下如何运动。这些问题具有收敛幂级数形式的全局解析解,比如,Karl F.Sundman证明n=3的情况,qaudong Wang证明n>3的情况。然而,Sundman级数和Wang级数收敛速度太慢,无法用于实际目的;因此,目前有必要通过数值分析以数值积分的形式来近似解,或者在某些情况下,采用经典三角级数近似。原子系统,例如原子、离子和分子,可以用量子N体问题来处理。在经典物理系统中,N体问题通常是指一个星系或一个星系团;行星系统,如恒星、行星及其卫星,也可以被视为N体系统。一些应用可以方便地用扰动理论来处理,其中系统被认为是一个两体问题加上导致偏离假设的无扰动两体轨道的附加力。


The problem is a plot device in the science fiction trilogy by Chinese author Cixin Liu, and its name has been used for both the first volume and the trilogy as a whole