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第159行： − 古典力学中的引力三体问题的量子力学类似物是氦原子，其中氦原子核和两个电子根据平方反 库仑相互作用而相互作用。像重力三体问题一样，氦原子不能精确地求解。+ + − 然而，在经典力学和量子力学中，除了平方反作用力之外，还存在非平凡的相互作用定律，这些定律确实导致了精确的三体分析解析。一种这样的模型包括谐波吸引和排斥反立方力的组合。[24]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，这导致了易于求解的线性微分方程组）。在这两个方面，它类似于具有库仑相互作用的（不溶性）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。+ − 使用广义相对论研究了引力三体问题。物理，相对论处理成为系统需要具有非常强大的引力场，如附近视界一个的黑洞。但是，相对论的问题比牛顿力学要困难得多，并且需要复杂的数值技术。即使是完整的两体问题（即对于任意质量比率），在广义相对论中也没有严格的解析解。+ − 三体问题是n体问题的特例，它描述了n个对象如何在一种物理力（例如重力）下运动。这些问题具有收敛的幂级数形式的全局解析解，这由Karl F. Sundman对于n = 3和由Qiudong Wang对于n > 3进行了证明。但是，Sundman和Wang系列收敛太慢，以至于在实际应用中毫无用处。因此，目前需要通过数值分析来近似解为数值积分或某些情况下的经典三角序列逼近。可以根据量子n体问题来处理原子，离子和分子等原子系统。在经典的物理系统中，n体问题通常是指星系或星系团。行星系统，例如恒星，行星及其卫星，也可以视为n体系统。某些应用程序容易受到干扰的影响 理论上，该系统被认为是一个两体问题，外加引起与假设的无扰动两体轨迹偏离的力。+ + +
无编辑摘要
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]
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]
氦原子是量子力学的模拟物中的模拟呜，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，还存在着一些非零的(nontrivial)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。该模型被认为是非零的(nontrivial)，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]
引力三体问题也用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.
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。