Editing Three-body problem (section)

===General solution===
[[File:3bodyproblem.gif|600px|thumb| While a system of 3 bodies interacting gravitationally is [[Chaos theory|chaotic]], a system of 3 bodies interacting [[Elastic force|elastically]] is not.{{clarify|date=April 2024}}]]

There is no general [[Closed-form expression|closed-form solution]] to the three-body problem.<ref name="PrincetonCompanion"/> In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.<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=https://www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem |date=8 March 2013 |access-date = 2013-04-04}}</ref>

However, in 1912 the [[Finland|Finnish]] [[Mathematics|mathematician]] [[Karl Fritiof Sundman]] proved that there exists an [[analytic solution]] to the three-body problem in the form of a [[Puiseux series]], specifically a [[power series]] in terms of powers of {{math|''t''<sup>1/3</sup>}}.<ref>[[June Barrow-Green|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> This series converges for all real {{mvar|t}}, except for initial conditions corresponding to zero [[angular momentum]]. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum 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 three-body problems. As is briefly discussed below, the only singularities in the three-body problem are binary collisions (collisions between two particles at an instant) and triple collisions (collisions between three particles at an instant).

Collisions of any number are somewhat improbable, since it has been shown that they correspond to a set of initial conditions of measure zero. But 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:
# Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as [[regularization (physics)|regularization]].
# Proving that triple collisions only occur when the angular momentum {{math|'''L'''}} vanishes. By restricting the initial data to {{math|'''L''' ≠ '''0'''}}, he removed all ''real'' singularities from the transformed equations for the three-body problem.
# Showing that if {{math|'''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 [[Cauchy]]'s [[existence theorem]] for differential equations, that there are no complex singularities in a strip (depending on the value of {{math|'''L'''}}) in the complex plane centered around the real axis (related to the [[Cauchy–Kovalevskaya theorem]]).
# Find a conformal transformation that maps this strip into the unit disc. For example, if {{math|1=''s'' = ''t''<sup>1/3</sup>}} (the new variable after the regularization) and if {{math|{{abs|ln ''s''}} ≤ ''β''}},{{clarify|date=December 2009|reason=define terms}} then this map is given by <math display="block">\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.

The corresponding series converges extremely 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 10<sup>{{val|8000000}}</sup> terms.<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>