Editing Three-body problem (section)

==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}_1 &= -G m_2 \frac{ \left( \mathbf{r}_1 - \mathbf{r}_2 \right) }{\ \left|\mathbf{r}_1 - \mathbf{r}_2 \right|^3 } - G m_3 \frac{ \left( \mathbf{r}_1 - \mathbf{r}_3 \right)}{\ \left|\mathbf{r}_1 - \mathbf{r}_3\right|^3}\ , \\
 \ddot\mathbf{r}_2 &= -G m_3 \frac{ \left( \mathbf{r}_2 - \mathbf{r}_3 \right) }{\ \left|\mathbf{r}_2 - \mathbf{r}_3 \right|^3 } - G m_1 \frac{ \left( \mathbf{r}_2 - \mathbf{r}_1 \right) }{\ \left|\mathbf{r}_2 - \mathbf{r}_1 \right|^3 }\ , \\
 \ddot\mathbf{r}_3 &= -G m_1 \frac{ \left( \mathbf{r}_3 - \mathbf{r}_1 \right) }{\ \left|\mathbf{r}_3 - \mathbf{r}_1 \right|^3 } - G m_2 \frac{ \left( \mathbf{r}_3 - \mathbf{r}_2 \right) }{\ \left|\mathbf{r}_3 - \mathbf{r}_2 \right|^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&nbsp;first order scalar differential equations."<ref>This explicit report on the vector presentation appears drawn, through generalisation, from the <math>\ \ddot\mathbf{r}_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}_i = \ldots\ </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 P_j = r_{ij}\ ,</math> and coordinates <math>\ q_{ij}\ </math> <math>\ (i,j = 1,2,3)\ </math> 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&nbsp;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{\mathrm d\ \mathbf{r}_i}{\mathrm d\ t} = \frac{\partial\ \mathcal{H} }{ \partial\ \mathbf{p}_i}\ , \qquad \frac{ \mathrm d\ \mathbf{p}_i }{ \mathrm d\ t } = -\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}{\left|\mathbf{r}_1 - \mathbf{r}_2\right|}\ -\ \frac{G m_2 m_3}{\left|\mathbf{r}_3 - \mathbf{r}_2\right|}\ -\ \frac{G m_3 m_1}{\left|\mathbf{r}_3 - \mathbf{r}_1\right|}\ +\ \frac{\left|\mathbf{p}_1\right|^2}{2m_1}\ +\ \frac{\left|\mathbf{p}_2\right|^2}{2m_2}\ +\ \frac{\left|\mathbf{p}_3\right|^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, m_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{\mathrm d^2 x}{\mathrm d\ t^2} = -m_1 \frac{(x - x_1)}{r_1^3} - m_2 \frac{(x - x_2)}{r_2^3}\ , \\
\frac{\mathrm d^2 y}{\mathrm d\ t^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 \equiv \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, if the two bodies are uniformly rotating, this time dependence can be removed through a transformation to their 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>