View source for Three-body problem

{{Short description|Physics problem related to laws of motion and gravity}}
{{about|the physics theory}}
[[File:Three-body Problem Animation with COM.gif|thumb|320px|Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. The [[center of mass]], in accordance with the [[law of conservation of momentum]], remains in place.]]
In [[physics]], specifically [[classical mechanics]], the '''three-body problem''' is to take the initial positions and velocities (or [[momentum|momenta]]) of three [[point mass]]es that orbit each other in space and calculate their subsequent trajectories using [[Newton's laws of motion]] and [[Newton's law of universal gravitation]].<ref name="PrincetonCompanion">{{cite encyclopedia
| last  = Barrow-Green
| first = June
| year  = 2008
| title = The Three-Body Problem
| editor-last1  = Gowers
| editor-first1 = Timothy
| editor-last2  = Barrow-Green
| editor-first2 = June
| editor-last3 = Leader
| editor-first3 = Imre
| encyclopedia = The Princeton Companion to Mathematics
| pages  = 726–728
| publisher = Princeton University Press
}}</ref> 

Unlike the [[two-body problem]], the three-body problem has no general [[closed-form expression|closed-form solution]], meaning there is no equation that always solves it.<ref name="PrincetonCompanion"/> When three bodies orbit each other, the resulting [[dynamical system]] is [[chaos theory|chaotic]] for most [[initial condition]]s. Because there are no solvable equations for most three-body systems, the only way to predict the motions of the bodies is to estimate them using [[numerical method]]s. 

The three-body problem is a special case of the [[n-body problem|{{mvar|n}}-body problem]]. Historically, the first specific three-body problem to receive extended study was the one involving the [[Earth]], the [[Moon]], and the [[Sun]].<ref name="first">{{cite web |url=http://www.wolframscience.com/reference/notes/972d |title=Historical Notes: Three-Body Problem |access-date=19 July 2017}}</ref> 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.

==Mathematical description==
The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions <math>\mathbf{r_i} = (x_i, y_i, z_i)</math> of three gravitationally interacting bodies with masses <math>m_i</math>:

<math display="block">\begin{align}
 \ddot\mathbf{r}_{\mathbf{1}} &= -G m_2 \frac{\mathbf{r_1} - \mathbf{r_2}}{|\mathbf{r_1} - \mathbf{r_2}|^3} - G m_3 \frac{\mathbf{r_1} - \mathbf{r_3}}{|\mathbf{r_1} - \mathbf{r_3}|^3}, \\
 \ddot\mathbf{r}_{\mathbf{2}} &= -G m_3 \frac{\mathbf{r_2} - \mathbf{r_3}}{|\mathbf{r_2} - \mathbf{r_3}|^3} - G m_1 \frac{\mathbf{r_2} - \mathbf{r_1}}{|\mathbf{r_2} - \mathbf{r_1}|^3}, \\
 \ddot\mathbf{r}_{\mathbf{3}} &= -G m_1 \frac{\mathbf{r_3} - \mathbf{r_1}}{|\mathbf{r_3} - \mathbf{r_1}|^3} - G m_2 \frac{\mathbf{r_3} - \mathbf{r_2}}{|\mathbf{r_3} - \mathbf{r_2}|^3}.
\end{align}</math>where <math>G</math> is the [[gravitational constant]]. As astronomer Juhan Frank describes, "These three second-order vector [[differential equation]]s are equivalent to 18 first order scalar differential equations."<ref>This explicit report on the vector presentation appears drawn, through generalisation, from the <math>\ddot\mathbf{r}_{\mathbf{i}}</math> expression presented by Cambridge-educated [[astronomer]] [[Juhan Frank]] of [[Louisiana State University|LSU]], as presented in his class notes for Physics 7221 in 2006, see {{Cite web |author=Frank, Juhan |date=October 11, 2006 |title=PHYS 7221 Special Lecture—The Three-Body Problem |url=http://www.phys.lsu.edu/faculty/gonzalez/Teaching/Phys7221/ThreeBodyProblem.pdf |publisher=Published by self, and [[Louisiana State University|LSU]] Department of Physics an Astronomy |format=class handout |quote=[Quoting] Just as in the two-body problem it is most convenient to work in the center-of-mass (CM) system with <math>\mathbf{x_i}</math> denoting the position of mass <math>\mathbf{m_i}</math>. The Newtonian equations of motion in this system are of the form <math>\ddot\mathbf{r}_{\mathbf{i}}</math> = ... |location=Baton Rouge, LA}}.{{better source|date=October 2024}}</ref>{{better source|date = October 2024}} As June Barrow-Green notes with regard to an alternative presentation, if <blockquote><math>P_i</math> represent three particles with masses <math>m_i</math>, distances <math>P_i</math><math>P_j</math> = <math>r_{ij}</math>, and coordinates <math>q_{ij}</math> (i,j = 1,2,3) in an inertial coordinate system ... the problem is described by nine second-order differential equations.<ref name="Barrow-Green1997">For a more general discussion of the presentation of these equations in non-vector formats not explicitly related to the presentation in text, see the authoritative {{cite book|first=June|last=Barrow-Green|author-link=June Barrow-Green|title=Poincaré and the Three Body Problem|title-link= Poincaré and the Three-Body Problem |year=1997|publisher=American Mathematical Society|isbn=978-0-8218-0367-7|pages=8–12|bibcode=1997ptbp.book.....B}}</ref>{{rp|8}}</blockquote> 

The problem can also be stated equivalently in the [[Hamiltonian formalism]], in which case it is described by a set of 18 first-order differential equations, one for each component of the positions <math>\mathbf{r_i}</math> and momenta <math>\mathbf{p_i}</math>:{{cn|date = October 2024}}<!--The Hamiltonian formulation is not in the preceding Frank citation, nor, explicitly/understandably, in pages 8-12 of the Barrow-Green citation. This is therefore WP:OR, and in need of a further citation.--><ref>For a related presentation of the Hamiltonian, which chooses units and presentation to simplify the maths, see Barrow-Green, p. 8, op. cit.</ref>

<math display="block">
\frac{d \mathbf{r_i}}{dt} = \frac{\partial \mathcal{H}}{\partial \mathbf{p_i}}, \qquad \frac{d\mathbf{p_i}}{dt} = -\frac{\partial \mathcal{H}}{\partial \mathbf{r_i}},
</math>

where <math>\mathcal{H}</math> is the [[Hamiltonian mechanics#Mathematical formalism|Hamiltonian]]:{{cn|date = October 2024}}<!--See note above regarding missing relevant sources.-->

<math display="block">
\mathcal{H} = -\frac{G m_1 m_2}{|\mathbf{r_1} - \mathbf{r_2}|}-\frac{G m_2 m_3}{|\mathbf{r_3} - \mathbf{r_2}|} -\frac{G m_3 m_1}{|\mathbf{r_3} - \mathbf{r_1}|} + \frac{\mathbf{p_1}^2}{2m_1} + \frac{\mathbf{p_2}^2}{2m_2} + \frac{\mathbf{p_3}^2}{2m_3}.
</math>

In this case, <math>\mathcal{H}</math> is simply the total energy of the system, gravitational plus kinetic.{{cn|date = October 2024}}<!--See note above regarding missing relevant sources.-->

===Restricted three-body problem{{anchor|Circular restricted three-body problem}}{{anchor|Circular}}===
{{more citations needed section|date = July 2023}}
[[File:Restricted Three-Body Problem - Energy Potential Analysis.png|thumb|300px|The circular restricted three-body problem{{clarify|date = July 2023}} is a valid approximation of elliptical orbits found in the [[Solar System]],{{citation needed|date = July 2023}} and this can be visualized as a combination of the potentials due to the gravity of the two primary bodies along with the centrifugal effect from their rotation ([[Coriolis effect]]s are dynamic and not shown). The [[Lagrange points]] can then be seen as the five places where the gradient on the resultant surface is zero, indicating that the forces are in balance there.{{citation needed|date = July 2023}}]]
<!--THIS IMAGE IS DRAWN FROM A GRADUATE DISSERTATION, THE LINKS TO WHICH HAVE SUBSEQUENTLY FAILED, AND SO THIS INFORMATION APPEARS TO BE UNTRACEABLE, UNVERIFIABLE, AND THEREFORE LARGELY USELESS.-->

In the ''restricted three-body problem'' formulation, in the description of Barrow-Green,<ref name="Barrow-Green1997"/>{{rp|11-14}}<blockquote>two... bodies revolve around their centre of mass in circular orbits under the influence of their mutual gravitational attraction, and... form a two body system... [whose] motion is known. A third body (generally known as a planetoid), assumed massless with respect to the other two, moves in the plane defined by the two revolving bodies and, while being gravitationally influenced by them, exerts no influence of its own.<ref name="Barrow-Green1997"/>{{rp|11}}</blockquote> Per Barrow-Green, "[t]he problem is then to ascertain the motion of the third body."<ref name="Barrow-Green1997"/>{{rp|11}}<!--<ref>Earlier editors thought it important, though being unrelated to the foregoing definition of problem, to call attention to [http://scienceworld.wolfram.com/physics/RestrictedThree-BodyProblem.html "Restricted Three-Body Problem"]. ''Eric Weisstein's World of Physics''. Wolfram Research.{{full|date = October 2024}}</ref>{{verification failed|reason=Source does not contain this claim.|date=May 2024}}-->

That is to say, 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.{{what|date = October 2024}}<!--This is not well-defined enough to be meaningful, compared to the quote, and we believe therefore adds nothing.--> (That is, it is useful to consider the [[effective potential]].{{what|date = October 2024}}{{says who|date = October 2024}}) With respect to a [[rotating reference frame]], the two co-orbiting bodies are stationary, and the third can be stationary as well at the [[Lagrangian points]], or move around them, for instance on a [[horseshoe orbit]].{{cn|date = October 2024}} 

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.<ref name="first2024">{{cite journal | author = Montgomery, Richard | url=https://www.scientificamerican.com/article/the-three-body-problem/ |title=The Three-Body Problem |journal=[[Scientific American]] |date=August 2019 | volume=321 | issue=2 | page=66 | doi=10.1038/scientificamerican0819-66 | pmid=39010603 |access-date=7 May 2024}}</ref>

Mathematically, the problem is stated as follows.{{cn|date=October 2024}}<!--Whose is this formulation?--> Let <math>m_{1,2}</math> be the masses of the two massive bodies, with (planar) coordinates <math>(x_1, y_1)</math> and <math>(x_2, y_2)</math>, and let <math>(x, y)</math> 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 <math>1</math>. Then, the motion of the planetoid is given by:{{cn|date=October 2024}} 

<math display="block">
\begin{align}
\frac{d^2 x}{dt^2} = -m_1 \frac{x - x_1}{r_1^3} - m_2 \frac{x - x_2}{r_2^3}, \\
\frac{d^2 y}{dt^2} = -m_1 \frac{y - y_1}{r_1^3} - m_2 \frac{y - y_2}{r_2^3},
\end{align}
</math>

where <math>r_i = \sqrt{(x - x_i)^2 + (y - y_i)^2}</math>.{{cn|date=October 2024}} In this form the equations of motion carry an explicit time dependence through the coordinates <math>x_i(t), y_i(t)</math>;{{cn|date=October 2024}} however, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.{{or|date = October 2024}}<ref>Note, the following source does not state that the "time dependence can be removed through a transformation to a rotating reference frame." For a related but distinct presentation of the restricted three-body problem—featuring the Jacobi integral for the "energy of <math>m_3</math> in the co-rotating (non-inertial) frame of the primaries"—see {{cite journal |title=An introduction to the classical three-body problem: From periodic solutions to instabilities and chaos |last1=Krishnaswami |first1=Govind S. |last2=Senapati |first2=Himalaya |journal=Resonance |volume=24 |pages=87–114, esp. p. 94f |year=2019 |publisher=Springer|doi=10.1007/s12045-019-0760-1 |arxiv=1901.07289 }}</ref>

==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>

==History==
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when [[Isaac Newton]] published his ''[[Philosophiæ Naturalis Principia Mathematica]],'' in which Newton attempted to figure out if any long term stability is possible especially for such a system like that of the [[Earth]], the [[Moon]], and [[Sun|the Sun]], after having solved the [[two-body problem]].<ref>{{Cite book |last=Musielak |first=Zdzislaw |url=https://books.google.com/books?id=D90tDwAAQBAJ&pg=PA3 |title=Three Body Dynamics and Its Applications to Exoplanets |last2=Quarles |first2=Billy |date=2017 |publisher=Springer International Publishing |isbn=978-3-319-58225-2 |series= |location= |pages=3 |language=en |doi=10.1007/978-3-319-58226-9}}</ref> Guided by major [[Renaissance]] astronomers [[Nicolaus Copernicus]], [[Tycho Brahe]] and [[Johannes Kepler]], Newton introduced later generations to the beginning of the gravitational three-body problem.<ref name=":1">{{Cite book |last=Valtonen |first=Mauri |author-link=Mauri Valtonen |url=https://books.google.com/books?id=4-wgDAAAQBAJ&pg=PA4 |title=The Three-body Problem from Pythagoras to Hawking |publisher=Springer |year=2016 |isbn=978-3-319-22726-9 |pages=4 |oclc=1171227640}}</ref> 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#Newton|lunar theory]], the motion of the Moon under the gravitational influence of Earth and the Sun.<ref>{{Cite book |last=Newton |first=Isaac |title=Philosophiæ naturalis principia mathematica |location=London |publisher=G. & J. Innys |date=1726 |url=https://lbezone.hkust.edu.hk/bib/b706487 |access-date=2022-10-05 |via=Hong Kong University of Science and Technology |doi=10.14711/spcol/b706487 }}</ref> Later, this problem was also applied to other planets' interactions with the Earth and the Sun.<ref name=":1" />

The physical problem was first addressed by [[Amerigo Vespucci]] and subsequently by [[Galileo Galilei]], as well as [[Simon Stevin]], but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions.<ref name=":1" /> Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil.<ref>{{Cite web |title=Amerigo Vespucci |url=https://www.biography.com/explorer/amerigo-vespucci |access-date=2022-10-05 |website=Biography |date=23 June 2021 |language=en-us}}</ref> It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the [[History of longitude#Problem of longitude|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 Earth.

[[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.<ref>The 1747 memoirs of both parties can be read in the volume of ''Histoires'' (including ''Mémoires'') of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):

: Clairaut: "On the System of the World, according to the principles of Universal Gravitation" (at pp.&nbsp;329–364); and
: d'Alembert: "General method for determining the orbits and the movements of all the planets, taking into account their mutual actions" (at pp.&nbsp;365–390).The peculiar dating is explained by a note printed on page 390 of the "Memoirs" section: "Even though the preceding memoirs, of Messrs. Clairaut and d'Alembert, were only read during the course of 1747, it was judged appropriate to publish them in the volume for this year" (i.e. the volume otherwise dedicated to the proceedings of 1745, but published in 1749).</ref> It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" ({{langx|fr|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.<ref>[[Jean le Rond d'Alembert]], in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol.&nbsp;2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp.&nbsp;329–312, at sec.&nbsp;VI, p.&nbsp;245.</ref>

From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P. F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a [[renormalization group]] [[limit cycle]] at the beginning of the 21st century.<ref name=":0">{{Cite journal |last1=Mohr |first1=R.F. |last2=Furnstahl |first2=R.J. |last3=Hammer |first3=H.-W. |last4=Perry |first4=R.J. |last5=Wilson |first5=K.G. |date=January 2006 |title=Precise numerical results for limit cycles in the quantum three-body problem |journal=[[Annals of Physics]] |volume=321 |issue=1 |pages=225–259 |doi=10.1016/j.aop.2005.10.002 |arxiv=nucl-th/0509076 |bibcode=2006AnPhy.321..225M |s2cid=119073191 |issn=0003-4916}}</ref> [[George William Hill]] worked on the restricted problem in the late 19th century with an application of motion of [[Venus]] and [[Mercury (planet)|Mercury]].<ref>[https://babel.hathitrust.org/cgi/pt?id=uc1.32106020676935&view=1up&seq=122&skin=2021 "Coplanar Motion of Two Planets, One Having a Zero Mass"]. [[Annals of Mathematics]], Vol. III, pp. 65–73, 1887.</ref>

At the beginning of the 20th century,  [[Karl F. Sundman|Karl Sundman]] approached the problem mathematically and systematically by providing a functional theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved the three-body problem. However, because there was not a qualitative enough solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved.<ref>{{Cite book |last=Barrow-Green |first=June |author-link=June Barrow-Green |title=Poincaré and the Three Body Problem |date=1996-10-29 |publisher=American Mathematical Society |isbn=978-0-8218-0367-7 |series=History of Mathematics |volume=11 |location=Providence, Rhode Island|doi=10.1090/hmath/011 |url=http://oro.open.ac.uk/57403/1/335423.pdf }}</ref> In the 1970s, implication to three-body from two-body forces had been discovered by [[Vitaly Efimov|V. Efimov]], which was named the [[Efimov effect]].<ref>{{Cite journal |last=Efimov |first=V. |date=1970-12-21 |title=Energy levels arising from resonant two-body forces in a three-body system |journal=Physics Letters B |language=en |volume=33 |issue=8 |pages=563–564 |doi=10.1016/0370-2693(70)90349-7 |bibcode=1970PhLB...33..563E |issn=0370-2693}}</ref>

In 2017,  [[Liao Shijun|Shijun Liao]] and Xiaoming Li applied a new strategy of numerical simulation for chaotic systems called the clean numerical simulation (CNS), with the use of a national supercomputer, to successfully gain 695 families of periodic solutions of the three-body system with equal mass.<ref>{{Cite journal |last1=Liao |first1=Shijun |last2=Li |first2=Xiaoming |date=2019-11-01 |title=On the periodic solutions of the three-body problem |url=https://academic.oup.com/nsr/article/6/6/1070/5537324 |journal=National Science Review |language=en |volume=6 |issue=6 |pages=1070–1071 |doi=10.1093/nsr/nwz102 |pmid=34691975 |pmc=8291409 |issn=2095-5138}}</ref>

In 2019, Breen et al. announced a fast [[neural network]] solver for the three-body problem, trained using a numerical integrator.<ref>{{cite journal |last1=Breen |first1=Philip G. |last2=Foley |first2=Christopher N. |last3=Boekholt |first3=Tjarda |last4=Portegies Zwart |first4=Simon |title=Newton versus the machine: Solving the chaotic three-body problem using deep neural networks |arxiv=1910.07291 |doi=10.1093/mnras/staa713 |journal=Monthly Notices of the Royal Astronomical Society |volume=494 |issue=2 |date=2020 |pages=2465–2470 |doi-access=free |s2cid=204734498}}</ref>

In September 2023, several possible solutions have been found to the problem according to reports.<ref name="SA-20230923">{{cite news |last=Watson |first=Claire |title=We Just Got 12,000 New Solutions to The Infamous Three-Body Problem |url=https://www.sciencealert.com/we-just-got-12000-new-solutions-to-the-infamous-three-body-problem |date=23 September 2023 |work=[[ScienceAlert]] |url-status=live |archiveurl=https://archive.today/20230924005541/https://www.sciencealert.com/we-just-got-12000-new-solutions-to-the-infamous-three-body-problem |archivedate=24 September 2023 |accessdate=23 September 2023 }}</ref><ref name="ARX-2023">{{cite journal |title=Three-body periodic collisionless equal-mass free-fall orbits revisited |date=2024 |arxiv=2308.16159 |last1=Hristov |first1=Ivan |last2=Hristova |first2=Radoslava |last3=Dmitrašinović |first3=Veljko |last4=Tanikawa |first4=Kiyotaka |journal=Celestial Mechanics and Dynamical Astronomy |volume=136 |issue=1 |doi=10.1007/s10569-023-10177-w |bibcode=2024CeMDA.136....7H }}</ref>

==Other problems involving three bodies==

The term "three-body problem" is sometimes used in the more general sense to refer to any physical problem involving the interaction of three bodies.

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.<ref>{{cite book | author=Griffiths, David J. | author-link=David J. Griffiths | title=Introduction to Quantum Mechanics (2nd ed.) | publisher=Prentice Hall |year=2004 |isbn=978-0-13-111892-8 | oclc=40251748 |page=311}}</ref>

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of [[Harmonic oscillator|harmonic attraction]] and a repulsive inverse-cube force.<ref name="Crandall1984">{{cite journal |last1=Crandall |first1= R. |last2=Whitnell |first2=R. |last3=Bettega |first3=R. |title= Exactly soluble two-electron atomic model |journal=American Journal of Physics |volume=52 |issue=5 |pages=438–442 |year=1984 |doi=10.1119/1.13650 |bibcode=1984AmJPh..52..438C}}</ref> 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.<ref name="Crandall1984"/><ref>{{cite journal |last=Calogero |first=F. |author-link=Francesco Calogero (physicist) |title=Solution of a Three-Body Problem in One Dimension |journal=Journal of Mathematical Physics |volume=10 |issue=12 |pages=2191–2196 |year=1969 |doi=10.1063/1.1664820 |bibcode= 1969JMP....10.2191C}}</ref>

Within the [[Two-dimensional point vortex gas|point vortex model]], the motion of [[Vortex|vortices]] in a two-dimensional ideal fluid is described by equations of motion that contain only first-order time derivatives. I.e. in contrast to Newtonian mechanics, it is the ''velocity'' and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still [[Integrable system|integrable]],<ref>{{Cite journal |last=Aref |first=Hassan |author-link=Hassan Aref |date=1979-03-01 |title=Motion of three vortices |url=https://aip.scitation.org/doi/10.1063/1.862605 |journal=The Physics of Fluids |volume=22 |issue=3 |pages=393–400 |doi=10.1063/1.862605 |bibcode=1979PhFl...22..393A |issn=0031-9171}}</ref> while at least four vortices are required to obtain chaotic behavior.<ref>{{Cite journal |last1=Aref |first1=Hassan |author-link1=Hassan Aref |last2=Pomphrey |first2=Neil |date=1980-08-18 |title=Integrable and chaotic motions of four vortices |journal=Physics Letters A |language=en |volume=78 |issue=4 |pages=297–300 |doi=10.1016/0375-9601(80)90375-8 |bibcode=1980PhLA...78..297A |issn=0375-9601}}</ref> One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.<ref>{{Cite journal |last1=Neufeld |first1=Z |last2=Tél |first2=T |date=1997-03-21 |title=The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices |url=https://iopscience.iop.org/article/10.1088/0305-4470/30/6/043 |journal=Journal of Physics A: Mathematical and General |volume=30 |issue=6 |pages=2263–2280 |doi=10.1088/0305-4470/30/6/043 |bibcode=1997JPhA...30.2263N |issn=0305-4470}}</ref>

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 [[Numerical relativity|sophisticated numerical techniques]] are required.
Even the full [[Two-body problem in general relativity|two-body problem]] (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.<ref>{{cite journal | last1=Musielak | first1=Z. E. | last2=Quarles | first2=B. | title=The three-body problem | journal=Reports on Progress in Physics | volume=77 | issue=6 | page=065901 | year=2014 | issn=0034-4885 | doi=10.1088/0034-4885/77/6/065901 | pmid=24913140| arxiv=1508.02312 | bibcode=2014RPPh...77f5901M | s2cid=38140668 }}</ref>

=={{mvar|n}}-body problem==
The three-body problem is a special case of the [[n-body problem|{{mvar|n}}-body problem]], which describes how {{mvar|n}} objects 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.&nbsp;Sundman]] for {{math|''n'' {{=}} 3}} and by [[Qiudong Wang]] for {{math|''n'' > 3}} (see [[n-body problem|{{mvar|n}}-body problem]] for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;<ref>[[Florin Diacu]]. [http://www.math.uvic.ca/faculty/diacu/diacuNbody.pdf "The Solution of the ''n''-body Problem"], ''[[The Mathematical Intelligencer]]'', 1996.</ref> 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|{{mvar|n}}-body simulation]]). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum {{mvar|n}}-body problem. Among classical physical systems, the {{mvar|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 {{mvar|n}}-body systems. Some applications are conveniently treated by [[perturbation (astronomy)|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.

==See also==
{{Portal|Mathematics|Physics}}
{{div col}}
* [[Few-body systems]]
* [[Galaxy formation and evolution]]
* [[Gravity assist]]
* [[Lagrange point]]
* [[Low-energy transfer]]
* [[Michael Minovitch]]
* [[n-body simulation|{{mvar|n}}-body simulation]]
* [[Symplectic integrator]]
* [[Sitnikov problem]]
* [[Two-body problem]]
* [[Synodic reference frame]] <!-- REDIRECT to stub section; needs help -->
* [[Triple star system]]
* [[The Three-Body Problem (novel)|''The Three-Body Problem'' (novel)]]
* [[3 Body Problem (TV series)|''3 Body Problem'' (TV series)]]
{{div col end}}

==References==
{{Reflist|2}}

==Further reading==
{{refbegin}}
* {{cite book |last=Aarseth |first=S. J. |title=Gravitational n-Body Simulations |date=2003 |publisher=Cambridge University Press |location=New York |isbn=978-0-521-43272-6 }}
* {{cite journal |last=Bagla |first=J. S. |title=Cosmological N-body simulation: Techniques, scope and status |date=2005 |journal=[[Current Science]] |volume=88 |pages=1088–1100 |arxiv=astro-ph/0411043 |bibcode = 2005CSci...88.1088B }}
* {{cite journal |last1=Chambers |first1=J. E. |last2=Wetherill |first2=G. W. |title=Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions |date=1998 |journal=[[Icarus (journal)|Icarus]] |volume=136 |issue=2 |pages=304–327 |doi=10.1006/icar.1998.6007 |bibcode = 1998Icar..136..304C |citeseerx=10.1.1.64.7797 }}
* {{cite journal |last1=Efstathiou |first1=G. |last2=Davis |first2=M. |last3=White |first3=S. D. M. |last4=Frenk |first4=C. S. |title=Numerical techniques for large cosmological N-body simulations |date=1985 |journal=[[Astrophysical Journal]] |volume= 57 |pages=241–260 |doi=10.1086/191003 |bibcode = 1985ApJS...57..241E |doi-access=free }}
* {{cite journal |last=Hulkower |first=Neal D. |title=The Zero Energy Three Body Problem |journal=[[Indiana University Mathematics Journal]] |volume=27 |issue=3 |date=1978 |pages=409–447 |doi= 10.1512/iumj.1978.27.27030|bibcode=1978IUMJ...27..409H |doi-access=free }}
* {{cite journal |last=Hulkower |first=Neal D. |title=Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem |journal=[[Celestial Mechanics (journal)|Celestial Mechanics]] |volume=21 |date=1980 |issue=1 |pages=37–41 |doi=10.1007/BF01230244 |bibcode = 1980CeMec..21...37H |s2cid=123404551 }}
* {{cite journal |last1=Li |first1=Xiaoming |last2=Liao |first2=Shijun |arxiv=1312.6796 |title=On the stability of the three classes of Newtonian three-body planar periodic orbits |date=2014 |journal=[[Science China Physics, Mechanics & Astronomy]] |volume=57 |issue=11 |pages= 2121–2126 |doi=10.1007/s11433-014-5563-5 |bibcode = 2014SCPMA..57.2121L |s2cid=73682020 }}
* {{cite journal |last1=Šuvakov |first1=Milovan |last2=Dmitrašinović |first2=V. |arxiv=1303.0181 |title=Three Classes of Newtonian Three-Body Planar Periodic Orbits |date=2013 |journal=[[Physical Review Letters]] |volume=110 |issue=10 |page=114301 |doi=10.1103/PhysRevLett.110.114301 |pmid=25166541 |bibcode = 2013PhRvL.110k4301S |s2cid=118554305 }}
{{refend}}

==External links==
* {{cite journal |first=Alain |last=Chenciner |author-link=Alain Chenciner |date=2007 |title=Three body problem |journal=[[Scholarpedia]] |volume=2 |number=10 |page=2111 |doi=10.4249/scholarpedia.2111 |bibcode=2007SchpJ...2.2111C |doi-access=free }}
* [https://www.livescience.com/physics-mathematics/the-3-body-problem-may-not-be-so-chaotic-after-all-new-study-suggests The '3-body problem' may not be so chaotic after all, new study suggests] (''Live Science,'' October 22, 2024)
* [https://www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem Physicists Discover a Whopping 13 New Solutions to Three-Body Problem] (''Science,'' March 8, 2013)

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