19
个编辑
更改
无编辑摘要
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.
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牛顿万有引力定律按照三点处质量体的初始位置和速度（或动量）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。与双体问题不同的是，三体问题不存在一般的闭式解，因为产生的动力系统对于大多数初始条件来说是混沌的，所以一般需要数值方法求解。
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.
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，第一个被拓展研究的特定三体问题是月球、地球和太阳构成的“三体”问题。从现代意义上讲，拓展的三体问题可以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==
==数学描述==
三体的数学表达式可以用三个质量为<math>m_i</math>相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：
三体的数学表达式可以用三个质量为<math>m_i</math>的相互作用的物体的矢量位置<math>\mathbf{r_i} = (x_i, y_i, z_i)</math>的牛顿运动方程来表示：
<math>\begin{align}
<math>\begin{align}
\end{align}</math>
\end{align}</math>
其中<math>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>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：
其中<math>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>\mathbf{r_i}</math>和动量<math>\mathbf{p_i}</math>的一个分量：
:<math>
:<math>
</math>
</math>
其中<math>\mathcal{H}</math>是 Hamiltonian 函数：
其中<math>\mathcal{H}</math>是哈密顿 Hamiltonian函数：
:<math>
:<math>
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.
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。
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天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.
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.
受限制的三体问题比完全的三体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-太阳的系统，这也是在三体问题的历史发展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
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>m_{1,2}</math>为两个小行星的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(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：
在数学的表述上，设<math>m_{1,2}</math>为两个大质量天体的质量，二维平面坐标<math>(x_1, y_1)</math>和<math>(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>
:<math>
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.)
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 {{math|''t''<sup>1/3</sup>}}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。
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 {{math|''t''<sup>1/3</sup>}}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).
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二元碰撞(两个粒子在瞬间的碰撞)和三元碰撞(三个粒子在瞬间的碰撞)，下面会进行简单的讨论。
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:
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:
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.
1. Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as regularization.
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。
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际奇点。
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。
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={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且如果{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则可由下式给出：
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={{math|''t''<sup>1/3</sup>}}（正则化后的新变量），并且{{math|{{abs|ln ''s''}} ≤ ''β''}}（需要证明），则映射可由下式给出：
:<math>\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math>
:<math>\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.
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]
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，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<sup>{{val|8000000}}</sup>项。
==特殊的求解方法==
==特殊的求解方法==
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.
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。
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种新的解族。
In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.
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。
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]
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]
2018年，Li和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===
===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]
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，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==
==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 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.
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。
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.
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Jean le Rond d'Alembert和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。这些都与他们的研究有关，在1740年代的巴黎，“三体问题”（法语：Problèmedes trois Corps）这个名字开始了被普遍使用。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。
建立了长期竞争关系的Jean le Rond d'Alembert和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。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。
氦原子是经典力学中引力三体问题的量子力学模拟，其中一个氦原子核和两个电子会产生反平方库仑相互作用，这种相互作用称为经典力学中的三体问题。就像重力三体问题一样，氦原子的三体问题没有精确解。
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]
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]
然而，在经典力学和量子力学中，除了平方反力外，还存在着一些非平凡相互作用规律，这些规律可以得到精确的解析解。有一种模型是由谐波吸引和排斥反立方体力的组合而成的。该模型被认为是非平凡的，因为它与一组包含奇异性的非线性微分方程组相关联（例如，与单独的谐波相互作用相比，该关联能够得到易于求解的线性微分方程组）。在这两种情况下，三体问题类似于具有库仑相互作用的（无法解释的）模型，因此，有人提出将其作为直观理解诸如氦原子之类的物理系统的工具。
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 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]
引力三体问题也被通过广义相对论进行了研究。在物理上，相对论性的处理在引力场非常强的系统中变得非常必要，比如在黑洞的视界附近。然而，相对论性问题比牛顿力学困难得多，需要复杂的数值技术。即使是完整的两体问题（即任意质量比）在广义相对论中也没有严格的解析解。
==n体问题==
==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.
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摄动理论来处理，其中系统被认为是一个两体问题加上导致偏离假设的未扰动两体轨道的附加力。
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。
==三体小说==
==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
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
三体问题是中国作家刘慈欣的科幻三部曲中有所提及的模拟游戏，也被用于第一卷和整个三部曲的书名。