Editing Three-body problem (section)

==Solutions==

===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>

===Special-case solutions===

<!--Linked to from [[Lagrangian Point]]-->


In 1767, [[Leonhard Euler]] found three families of periodic solutions in which the three masses are [[collinear]] at each instant. 

In 1772, [[Joseph Louis Lagrange|Lagrange]] found a family of solutions in which the three masses form an equilateral triangle at each instant. Together with Euler's collinear solutions, these solutions form the [[central configuration]]s for the three-body problem. These solutions are valid for any mass ratios, and the masses move on [[Kepler orbit|Keplerian ellipses]]. These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the [[circular restricted three-body problem]], these solutions, viewed in a frame rotating with the primaries, become points called [[Lagrangian point]]s and labeled L<sub>1</sub>, L<sub>2</sub>, L<sub>3</sub>, L<sub>4</sub>, and L<sub>5</sub>, with L<sub>4</sub> and L<sub>5</sub> being symmetric instances of Lagrange's solution.

In work summarized in 1892–1899, [[Henri Poincaré]] established the existence of an infinite number of periodic solutions to the restricted three-body problem, together with techniques for continuing these solutions into the general three-body problem.

In 1893, Meissel stated what is now called the Pythagorean three-body problem: three masses in the ratio 3:4:5 are placed at rest at the vertices of a [[Special right triangle|3:4:5 right triangle]], with the heaviest body at the right angle and the lightest at the smaller acute angle. Burrau<ref name="Burrau">{{Cite journal |author1=Burrau |title=Numerische Berechnung eines Spezialfalles des Dreikörperproblems|journal=Astronomische Nachrichten |volume=195 |issue=6 |pages=113–118 |date=1913 |bibcode=1913AN....195..113B |doi=10.1002/asna.19131950602|url=https://zenodo.org/record/1424886}}</ref> further investigated this problem in 1913. In 1967 [[Victor Szebehely]] and [[C. Frederick Peters]] established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution.<ref>{{cite journal |last1=Victor Szebehely |last2=C. Frederick Peters |title=Complete Solution of a General Problem of Three Bodies |journal=Astronomical Journal |volume=72 |page=876 |date=1967 |doi=10.1086/110355 |bibcode=1967AJ.....72..876S |doi-access=free }}</ref>

[[File:Three body problem figure-8 orbit animation.gif|400px|thumb|An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259<ref>Here the gravitational constant ''G'' has been set to 1, and the initial conditions are '''r'''<sub>1</sub>(0) = -'''r'''<sub>3</sub>(0) = (-0.97000436, 0.24308753); '''r'''<sub>2</sub>(0) = (0,0); '''v'''<sub>1</sub>(0) = '''v'''<sub>3</sub>(0) = (0.4662036850, 0.4323657300); '''v'''<sub>2</sub>(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000).</ref>]]
[[File:5 4 800 36 downscaled.gif|500px|thumb|20 examples of periodic solutions to the three-body problem]]
In the 1970s, [[Michel Hénon]] and [[Roger A. Broucke]] each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this family, the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path.<ref name="TBG">{{cite web |author1=Šuvakov, M. |author2=Dmitrašinović, V. |title=Three-body Gallery |url=http://suki.ipb.ac.rs/3body/ |access-date=12 August 2015}}</ref>

In 1993, physicist [[Cris Moore]] at the [[Santa Fe Institute]] found a zero angular momentum solution with three equal masses moving around a figure-eight shape.<ref>{{cite journal
 | last = Moore
 | first = Cristopher
 | bibcode = 1993PhRvL..70.3675M
 | doi = 10.1103/PhysRevLett.70.3675
 | issue = 24
 | journal = Physical Review Letters
 | pages = 3675–3679
 | pmid = 10053934
 | title = Braids in classical dynamics
 | url = http://tuvalu.santafe.edu/~moore/braids-prl.pdf
 | volume = 70
 | year = 1993
 | access-date = 2016-01-01
 | archive-date = 2018-10-08
 | archive-url = https://web.archive.org/web/20181008213647/http://tuvalu.santafe.edu/~moore/braids-prl.pdf
 }}</ref> In 2000, mathematicians [[Alain Chenciner]] and Richard Montgomery proved its formal existence.<ref>{{cite journal|author=Chenciner, Alain|author2=Montgomery, Richard|title=A remarkable periodic solution of the three-body problem in the case of equal masses|journal=Annals of Mathematics |series=Second Series|volume=152|issue=3|year=2000|pages=881–902|doi=10.2307/2661357|arxiv=math/0011268|jstor=2661357|bibcode=2000math.....11268C|s2cid=10024592}}</ref><ref>{{cite journal
 | last = Montgomery | first = Richard
 | volume = 48
 | journal = Notices of the American Mathematical Society
 | pages = 471–481
 | title = A new solution to the three-body problem
 | url = https://www.ams.org/notices/200105/fea-montgomery.pdf
 | year = 2001}}</ref> The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which makes it possible for such orbits to be observed in the physical universe. But it has been argued that this is unlikely since the domain of stability is small. For instance, the probability of a binary–binary [[scattering]] event{{clarify|date=April 2020}} resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.<ref>{{cite journal
 | last = Heggie | first = Douglas C.
 | author-link = Douglas C. Heggie
 | doi = 10.1046/j.1365-8711.2000.04027.x
 | volume = 318
 | issue = 4
 | journal = Monthly Notices of the Royal Astronomical Society
 | pages = L61–L63
 | title = A new outcome of binary–binary scattering
 | year = 2000| doi-access = free
 | arxiv = astro-ph/9604016
 | bibcode = 2000MNRAS.318L..61H
 }}</ref>

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.<ref name=13solutions/><ref name="TBG"/>

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<ref>{{cite journal |last1=Hudomal |first1=Ana |title=New periodic solutions to the three-body problem and gravitational waves |journal=Master of Science Thesis at the Faculty of Physics, Belgrade University |date=October 2015 |url=http://www.scl.rs/theses/msc_ahudomal.pdf |access-date=5 February 2019}}</ref>

In 2017, researchers Xiaoming Li and [[Shijun Liao]] found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.<ref>{{cite journal |last1=Li |first1=Xiaoming |last2=Liao |first2=Shijun |title=More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits |journal=Science China Physics, Mechanics & Astronomy |date=December 2017 |volume=60 |issue=12 |page=129511 |doi=10.1007/s11433-017-9078-5 |arxiv=1705.00527 |issn=1674-7348|bibcode=2017SCPMA..60l9511L |s2cid=84838204 }}</ref> This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.<ref>{{cite journal |last1=Li |first1=Xiaoming |last2=Jing |first2=Yipeng |last3=Liao |first3=Shijun |title=The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum |journal=Publications of the Astronomical Society of Japan |volume=70 |issue=4 |date=August 2018 |article-number=64 |arxiv=1709.04775 |doi=10.1093/pasj/psy057 |doi-access=free}}</ref>

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem.<ref>{{cite journal |last1=Li |first1=Xiaoming |last2=Liao |first2=Shijun |title=Collisionless periodic orbits in the free-fall three-body problem |journal=New Astronomy |volume=70 |pages=22–26 |year=2019 |arxiv=1805.07980 |doi=10.1016/j.newast.2019.01.003 |bibcode=2019NewA...70...22L |s2cid=89615142 }}</ref> The free-fall formulation 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 forward and backward along an open "track".

In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, and found 12,409 distinct solutions.<ref>{{Cite journal |title=Three-body periodic collisionless equal-mass free-fall orbits revisited  |last1=Hristov |first1=Ivan |last2=Hristova |first2=Radoslava |last3=Dmitrašinović |first3=Veljko |last4=Tanikawa |first4=Kiyotaka |journal=Celestial Mechanics and Dynamical Astronomy |date=2024 |volume=136 |issue=1 |page=7 |doi=10.1007/s10569-023-10177-w |arxiv=2308.16159 |bibcode=2024CeMDA.136....7H }}</ref>

=== Numerical approaches ===
Using a computer, the problem may be solved to arbitrarily high precision using [[numerical integration]]. There have been attempts of creating computer programs that [[Numerical analysis|numerically solve]] the three-body problem (and by extension, the [[n-body problem]]) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as [[special relativity]].<ref>{{Cite web |title=3body simulator |url=https://3body.hk/ |access-date=2022-11-17 |website=3body simulator |language=en |archive-date=2022-11-17 |archive-url=https://web.archive.org/web/20221117041052/https://3body.hk/ |url-status=dead }}</ref> In addition, using the theory of [[random walks]], an approximate [[probability]] of different outcomes may be computed.<ref>{{cite news |last1=Technion |title=A Centuries-Old Physics Mystery? Solved |url=https://scitechdaily.com/a-centuries-old-physics-mystery-solved/ |access-date=12 October 2021 |work=SciTechDaily |publisher=[[SciTech (magazine)|SciTech]] |date=6 October 2021}}</ref><ref>{{cite journal |last1=Ginat |first1=Yonadav Barry |last2=Perets |first2=Hagai B. |title=Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters |journal=[[Physical Review]] |date=23 July 2021 |volume=11 |issue=3 |page=031020 |doi=10.1103/PhysRevX.11.031020|arxiv=2011.00010 |bibcode=2021PhRvX..11c1020G |s2cid=235485570 |url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.11.031020 |access-date=12 October 2021}}</ref>