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第72行： + − 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 卡尔·弗里蒂奥夫·桑德曼 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). − 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 − 第107行：
第93行： − This finishes the proof of Sundman's theorem.+ − 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] − + + +
→求解
==求解==
==求解==
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。<ref name="PrincetonCompanion"/> 此外，除特殊情况，三个物体的运动一般是不重复的。<ref name=13solutions>{{cite journal |first=Jon |last=Cartwright |title=Physicists Discover a Whopping 13 New Solutions to Three-Body Problem | journal=Science Now |url=http://www.sciencemag.org/news/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problem |date=8 March 2013 |access-date = 2013-04-04}}</ref>
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。<ref>Barrow-Green, J. (2010). [http://oro.open.ac.uk/22440/2/Sundman_final.pdf The dramatic episode of Sundman], Historia Mathematica 37, pp. 164–203.</ref>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。
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。
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。
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:
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. 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。
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。
:<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>
上述即为完整的Sundman定律的证明。
上述即为完整的Sundman定律的证明。
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<sup>{{val|8000000}}</sup>项。
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<sup>{{val|8000000}}</sup>项。<ref>{{cite journal |last=Beloriszky |first=D. |year=1930 |title=Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps |journal=Bulletin Astronomique |volume=6 |series=Série 2 |pages=417–434|bibcode=1930BuAst...6..417B }}</ref>
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==特殊的求解方法==
==特殊的求解方法==