# 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.

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.

## 数学描述

\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} }

$\displaystyle{ \frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}}, }$

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

## 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.

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

\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} }

## 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]

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.)

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:

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：

$\displaystyle{ \sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}. }$

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]

## 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， 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。

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。

### 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]

## 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.

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.

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.

## 其他涉及三体的问题

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]

## N体问题

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.

## 三体小说

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