Three-body problem

This article is about the physics theory. For other uses, see Three-body problem (disambiguation).

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 momenta) of three point masses that orbit each other in space and calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.^[1]

Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no equation that always solves it.^[1] When three bodies orbit each other, the resulting dynamical system is chaotic for most initial conditions. 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 methods.

The three-body problem is a special case of the 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.^[2] 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 ${\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})}$ of three gravitationally interacting bodies with masses ${\displaystyle m_{i}}$:

$${\displaystyle {\begin{aligned}{\ddot {\mathbf {r} }}_{\mathbf {1} }&=-Gm_{2}{\frac {\mathbf {r_{1}} -\mathbf {r_{2}} }{|\mathbf {r_{1}} -\mathbf {r_{2}} |^{3}}}-Gm_{3}{\frac {\mathbf {r_{1}} -\mathbf {r_{3}} }{|\mathbf {r_{1}} -\mathbf {r_{3}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {2} }&=-Gm_{3}{\frac {\mathbf {r_{2}} -\mathbf {r_{3}} }{|\mathbf {r_{2}} -\mathbf {r_{3}} |^{3}}}-Gm_{1}{\frac {\mathbf {r_{2}} -\mathbf {r_{1}} }{|\mathbf {r_{2}} -\mathbf {r_{1}} |^{3}}},\\{\ddot {\mathbf {r} }}_{\mathbf {3} }&=-Gm_{1}{\frac {\mathbf {r_{3}} -\mathbf {r_{1}} }{|\mathbf {r_{3}} -\mathbf {r_{1}} |^{3}}}-Gm_{2}{\frac {\mathbf {r_{3}} -\mathbf {r_{2}} }{|\mathbf {r_{3}} -\mathbf {r_{2}} |^{3}}}.\end{aligned}}}$$where ${\displaystyle G}$ is the gravitational constant. As astronomer Juhan Frank describes, "These three second-order vector differential equations are equivalent to 18 first order scalar differential equations."^{[3][better source needed]} As June Barrow-Green notes with regard to an alternative presentation, if

${\displaystyle P_{i}}$ represent three particles with masses ${\displaystyle m_{i}}$, distances ${\displaystyle P_{i}}$${\displaystyle P_{j}}$ = ${\displaystyle r_{ij}}$, and coordinates ${\displaystyle q_{ij}}$ (i,j = 1,2,3) in an inertial coordinate system ... the problem is described by nine second-order differential equations.^[4]: 8

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 ${\displaystyle \mathbf {r_{i}} }$ and momenta ${\displaystyle \mathbf {p_{i}} }$:^{[citation needed][5]}

$${\displaystyle {\frac {d\mathbf {r_{i}} }{dt}}={\frac {\partial {\mathcal {H}}}{\partial \mathbf {p_{i}} }},\qquad {\frac {d\mathbf {p_{i}} }{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \mathbf {r_{i}} }},}$$

where ${\displaystyle {\mathcal {H}}}$ is the Hamiltonian:^{[citation needed]}

$${\displaystyle {\mathcal {H}}=-{\frac {Gm_{1}m_{2}}{|\mathbf {r_{1}} -\mathbf {r_{2}} |}}-{\frac {Gm_{2}m_{3}}{|\mathbf {r_{3}} -\mathbf {r_{2}} |}}-{\frac {Gm_{3}m_{1}}{|\mathbf {r_{3}} -\mathbf {r_{1}} |}}+{\frac {\mathbf {p_{1}} ^{2}}{2m_{1}}}+{\frac {\mathbf {p_{2}} ^{2}}{2m_{2}}}+{\frac {\mathbf {p_{3}} ^{2}}{2m_{3}}}.}$$

In this case, ${\displaystyle {\mathcal {H}}}$ is simply the total energy of the system, gravitational plus kinetic.^{[citation needed]}

Restricted three-body problem

The circular restricted three-body problem^{[clarification needed]} is a valid approximation of elliptical orbits found in the Solar System,^{[citation needed]} 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 effects 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]}

In the restricted three-body problem formulation, in the description of Barrow-Green,^{[4]: 11–14}

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.^[4]: 11

Per Barrow-Green, "[t]he problem is then to ascertain the motion of the third body."^[4]: 11

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.^{[clarification needed]} (That is, it is useful to consider the effective potential.^{[clarification needed][according to whom?]}) 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.^{[citation needed]}

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.^[6]

Mathematically, the problem is stated as follows.^{[citation needed]} Let ${\displaystyle m_{1,2}}$ be the masses of the two massive bodies, with (planar) coordinates ${\displaystyle (x_{1},y_{1})}$ and ${\displaystyle (x_{2},y_{2})}$, and let ${\displaystyle (x,y)}$ 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 ${\displaystyle 1}$. Then, the motion of the planetoid is given by:^{[citation needed]}

$${\displaystyle {\begin{aligned}{\frac {d^{2}x}{dt^{2}}}=-m_{1}{\frac {x-x_{1}}{r_{1}^{3}}}-m_{2}{\frac {x-x_{2}}{r_{2}^{3}}},\\{\frac {d^{2}y}{dt^{2}}}=-m_{1}{\frac {y-y_{1}}{r_{1}^{3}}}-m_{2}{\frac {y-y_{2}}{r_{2}^{3}}},\end{aligned}}}$$

where ${\displaystyle r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}}$.^{[citation needed]} In this form the equations of motion carry an explicit time dependence through the coordinates ${\displaystyle x_{i}(t),y_{i}(t)}$;^{[citation needed]} however, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis.^{[original research?][7]}

Solutions

General solution

While a system of 3 bodies interacting gravitationally is chaotic, a system of 3 bodies interacting elastically is not.^{[clarification needed]}

There is no general closed-form solution to the three-body problem.^[1] 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.^[8]

However, in 1912 the Finnish 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 t^1/3.^[9] This series converges for all real 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:

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 three-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 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 (related to the Cauchy–Kovalevskaya theorem).
4. Find a conformal transformation that maps this strip into the unit disc. For example, if s = t^1/3 (the new variable after the regularization) and if |ln s| ≤ β,^{[clarification needed]} then this map is given by $${\displaystyle \sigma ={\frac {e^{\frac {\pi s}{2\beta }}-1}{e^{\frac {\pi s}{2\beta }}+1}}.}$$

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 10^8000000 terms.^[10]

Special-case solutions

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

In 1772, 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 configurations for the three-body problem. These solutions are valid for any mass ratios, and the masses move on 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 points and labeled L₁, L₂, L₃, L₄, and L₅, with L₄ and L₅ 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 3:4:5 right triangle, with the heaviest body at the right angle and the lightest at the smaller acute angle. Burrau^[11] 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.^[12]

An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259^[13]

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.^[14]

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.^[15] In 2000, mathematicians Alain Chenciner and Richard Montgomery proved its formal existence.^[16][17] 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^{[clarification needed]} resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.^[18]

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.^[8][14]

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.^[19]

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.^[20] This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.^[21]

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem.^[22] 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.^[23]

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 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.^[24] In addition, using the theory of random walks, an approximate probability of different outcomes may be computed.^[25][26]

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 the Sun, after having solved the two-body problem.^[27] 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.^[28] 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, the motion of the Moon under the gravitational influence of Earth and the Sun.^[29] Later, this problem was also applied to other planets' interactions with the Earth and the Sun.^[28]

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.^[28] Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil.^[30] It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the 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.^[31] It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (French: 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.^[32]

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.^[33] George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.^[34]

At the beginning of the 20th century, 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.^[35] In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov, which was named the Efimov effect.^[36]

In 2017, 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.^[37]

In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator.^[38]

In September 2023, several possible solutions have been found to the problem according to reports.^[39][40]

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.^[41]

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 attraction and a repulsive inverse-cube force.^[42] 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.^[42][43]

Within the point vortex model, the motion of 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,^[44] while at least four vortices are required to obtain chaotic behavior.^[45] 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.^[46]

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 sophisticated numerical techniques are required. Even the full two-body problem (i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity.^[47]

n-body problem

The three-body problem is a special case of the n-body problem, which describes how 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. Sundman for n = 3 and by Qiudong Wang for n > 3 (see n-body problem for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;^[48] 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). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum n-body problem. Among classical physical systems, the 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 n-body systems. Some applications are conveniently treated by 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

• Few-body systems
• Galaxy formation and evolution
• Gravity assist
• Lagrange point
• Low-energy transfer
• Michael Minovitch
• n-body simulation
• Symplectic integrator
• Sitnikov problem
• Two-body problem
• Synodic reference frame
• Triple star system
• The Three-Body Problem (novel)
• 3 Body Problem (TV series)

References

1. Barrow-Green, June (2008). "The Three-Body Problem". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.). The Princeton Companion to Mathematics. Princeton University Press. pp. 726–728.
2. "Historical Notes: Three-Body Problem". Retrieved 19 July 2017.
3. This explicit report on the vector presentation appears drawn, through generalisation, from the ${\displaystyle {\ddot {\mathbf {r} }}_{\mathbf {i} }}$ expression presented by Cambridge-educated astronomer Juhan Frank of LSU, as presented in his class notes for Physics 7221 in 2006, see Frank, Juhan (October 11, 2006). "PHYS 7221 Special Lecture—The Three-Body Problem" (class handout). Baton Rouge, LA: Published by self, and LSU Department of Physics an Astronomy. [Quoting] Just as in the two-body problem it is most convenient to work in the center-of-mass (CM) system with ${\displaystyle \mathbf {x_{i}} }$ denoting the position of mass ${\displaystyle \mathbf {m_{i}} }$. The Newtonian equations of motion in this system are of the form ${\displaystyle {\ddot {\mathbf {r} }}_{\mathbf {i} }}$ = ....^{[better source needed]}
4. 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 Barrow-Green, June (1997). Poincaré and the Three Body Problem. American Mathematical Society. pp. 8–12. Bibcode:1997ptbp.book.....B. ISBN 978-0-8218-0367-7.
5. For a related presentation of the Hamiltonian, which chooses units and presentation to simplify the maths, see Barrow-Green, p. 8, op. cit.
6. Montgomery, Richard (August 2019). "The Three-Body Problem". Scientific American. 321 (2): 66. doi:10.1038/scientificamerican0819-66. PMID 39010603. Retrieved 7 May 2024.
7. 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 ${\displaystyle m_{3}}$ in the co-rotating (non-inertial) frame of the primaries"—see Krishnaswami, Govind S.; Senapati, Himalaya (2019). "An introduction to the classical three-body problem: From periodic solutions to instabilities and chaos". Resonance. 24. Springer: 87–114, esp. p. 94f. arXiv:1901.07289. doi:10.1007/s12045-019-0760-1.
8. Cartwright, Jon (8 March 2013). "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem". Science Now. Retrieved 2013-04-04.
9. Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.
10. Beloriszky, D. (1930). "Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps". Bulletin Astronomique. Série 2. 6: 417–434. Bibcode:1930BuAst...6..417B.
11. Burrau (1913). "Numerische Berechnung eines Spezialfalles des Dreikörperproblems". Astronomische Nachrichten. 195 (6): 113–118. Bibcode:1913AN....195..113B. doi:10.1002/asna.19131950602.
12. Victor Szebehely; C. Frederick Peters (1967). "Complete Solution of a General Problem of Three Bodies". Astronomical Journal. 72: 876. Bibcode:1967AJ.....72..876S. doi:10.1086/110355.
13. Here the gravitational constant G has been set to 1, and the initial conditions are r₁(0) = -r₃(0) = (-0.97000436, 0.24308753); r₂(0) = (0,0); v₁(0) = v₃(0) = (0.4662036850, 0.4323657300); v₂(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000).
14. Šuvakov, M.; Dmitrašinović, V. "Three-body Gallery". Retrieved 12 August 2015.
15. Moore, Cristopher (1993). "Braids in classical dynamics" (PDF). Physical Review Letters. 70 (24): 3675–3679. Bibcode:1993PhRvL..70.3675M. doi:10.1103/PhysRevLett.70.3675. PMID 10053934. Archived from the original (PDF) on 2018-10-08. Retrieved 2016-01-01.
16. Chenciner, Alain; Montgomery, Richard (2000). "A remarkable periodic solution of the three-body problem in the case of equal masses". Annals of Mathematics. Second Series. 152 (3): 881–902. arXiv:math/0011268. Bibcode:2000math.....11268C. doi:10.2307/2661357. JSTOR 2661357. S2CID 10024592.
17. Montgomery, Richard (2001). "A new solution to the three-body problem" (PDF). Notices of the American Mathematical Society. 48: 471–481.
18. Heggie, Douglas C. (2000). "A new outcome of binary–binary scattering". Monthly Notices of the Royal Astronomical Society. 318 (4): L61 – L63. arXiv:astro-ph/9604016. Bibcode:2000MNRAS.318L..61H. doi:10.1046/j.1365-8711.2000.04027.x.
19. Hudomal, Ana (October 2015). "New periodic solutions to the three-body problem and gravitational waves" (PDF). Master of Science Thesis at the Faculty of Physics, Belgrade University. Retrieved 5 February 2019.
20. Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits". Science China Physics, Mechanics & Astronomy. 60 (12): 129511. arXiv:1705.00527. Bibcode:2017SCPMA..60l9511L. doi:10.1007/s11433-017-9078-5. ISSN 1674-7348. S2CID 84838204.
21. Li, Xiaoming; Jing, Yipeng; Liao, Shijun (August 2018). "The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum". Publications of the Astronomical Society of Japan. 70 (4) 64. arXiv:1709.04775. doi:10.1093/pasj/psy057.
22. Li, Xiaoming; Liao, Shijun (2019). "Collisionless periodic orbits in the free-fall three-body problem". New Astronomy. 70: 22–26. arXiv:1805.07980. Bibcode:2019NewA...70...22L. doi:10.1016/j.newast.2019.01.003. S2CID 89615142.
23. Hristov, Ivan; Hristova, Radoslava; Dmitrašinović, Veljko; Tanikawa, Kiyotaka (2024). "Three-body periodic collisionless equal-mass free-fall orbits revisited". Celestial Mechanics and Dynamical Astronomy. 136 (1): 7. arXiv:2308.16159. Bibcode:2024CeMDA.136....7H. doi:10.1007/s10569-023-10177-w.
24. "3body simulator". 3body simulator. Archived from the original on 2022-11-17. Retrieved 2022-11-17.
25. Technion (6 October 2021). "A Centuries-Old Physics Mystery? Solved". SciTechDaily. SciTech. Retrieved 12 October 2021.
26. Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021). "Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters". Physical Review. 11 (3): 031020. arXiv:2011.00010. Bibcode:2021PhRvX..11c1020G. doi:10.1103/PhysRevX.11.031020. S2CID 235485570. Retrieved 12 October 2021.
27. Musielak, Zdzislaw; Quarles, Billy (2017). Three Body Dynamics and Its Applications to Exoplanets. Springer International Publishing. p. 3. doi:10.1007/978-3-319-58226-9. ISBN 978-3-319-58225-2.
28. Valtonen, Mauri (2016). The Three-body Problem from Pythagoras to Hawking. Springer. p. 4. ISBN 978-3-319-22726-9. OCLC 1171227640.
29. Newton, Isaac (1726). Philosophiæ naturalis principia mathematica. London: G. & J. Innys. doi:10.14711/spcol/b706487. Retrieved 2022-10-05 – via Hong Kong University of Science and Technology.
30. "Amerigo Vespucci". Biography. 23 June 2021. Retrieved 2022-10-05.
31. 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. 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. 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).

32. 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. 2, Paris 1761, Quatorzième Mémoire ("Réflexions sur le Problème des trois Corps, avec de Nouvelles Tables de la Lune ...") pp. 329–312, at sec. VI, p. 245.
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Further reading

• Aarseth, S. J. (2003). Gravitational n-Body Simulations. New York: Cambridge University Press. ISBN 978-0-521-43272-6.
• Bagla, J. S. (2005). "Cosmological N-body simulation: Techniques, scope and status". Current Science. 88: 1088–1100. arXiv:astro-ph/0411043. Bibcode:2005CSci...88.1088B.
• Chambers, J. E.; Wetherill, G. W. (1998). "Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions". Icarus. 136 (2): 304–327. Bibcode:1998Icar..136..304C. CiteSeerX 10.1.1.64.7797. doi:10.1006/icar.1998.6007.
• Efstathiou, G.; Davis, M.; White, S. D. M.; Frenk, C. S. (1985). "Numerical techniques for large cosmological N-body simulations". Astrophysical Journal. 57: 241–260. Bibcode:1985ApJS...57..241E. doi:10.1086/191003.
• Hulkower, Neal D. (1978). "The Zero Energy Three Body Problem". Indiana University Mathematics Journal. 27 (3): 409–447. Bibcode:1978IUMJ...27..409H. doi:10.1512/iumj.1978.27.27030.
• Hulkower, Neal D. (1980). "Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem". Celestial Mechanics. 21 (1): 37–41. Bibcode:1980CeMec..21...37H. doi:10.1007/BF01230244. S2CID 123404551.
• Li, Xiaoming; Liao, Shijun (2014). "On the stability of the three classes of Newtonian three-body planar periodic orbits". Science China Physics, Mechanics & Astronomy. 57 (11): 2121–2126. arXiv:1312.6796. Bibcode:2014SCPMA..57.2121L. doi:10.1007/s11433-014-5563-5. S2CID 73682020.
• Šuvakov, Milovan; Dmitrašinović, V. (2013). "Three Classes of Newtonian Three-Body Planar Periodic Orbits". Physical Review Letters. 110 (10): 114301. arXiv:1303.0181. Bibcode:2013PhRvL.110k4301S. doi:10.1103/PhysRevLett.110.114301. PMID 25166541. S2CID 118554305.

External links

• Chenciner, Alain (2007). "Three body problem". Scholarpedia. 2 (10): 2111. Bibcode:2007SchpJ...2.2111C. doi:10.4249/scholarpedia.2111.
• The '3-body problem' may not be so chaotic after all, new study suggests (Live Science, October 22, 2024)
• Physicists Discover a Whopping 13 New Solutions to Three-Body Problem (Science, March 8, 2013)