Three-body problem - Knowledge (XXG)

989: 292: 984:{\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}}} 3279: 1731: 43: 5948: 3265: 1636: 4472:, 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. 5940: 2967:, 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. Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil. It became of technical importance in the 1720s, as an accurate solution would be applicable to navigation, specifically for the 1275: 1678: 2802: 2267: 1238: 5173: 2794: 1631:{\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}}}.} 1963: 2948:, 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 2993:, 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. It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" ( 1754:, a body of negligible mass (the "planetoid") moves under the influence of two massive bodies. Having negligible mass, the planetoid exerts force on the two massive bodies that may be neglected; therefore the resulting system can be analyzed and described as a two-body motion problem. With respect to a 4457:

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

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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

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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,

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and Richard Montgomery proved its formal existence. 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

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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

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and a repulsive inverse-cube force. 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

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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

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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

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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.

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

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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

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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.

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to the three-body problem. 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.

3116:, while at least four vortices are required to obtain chaotic behavior. 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. 2438: 2696: 2262:{\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}}} 3091:

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

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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.

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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).

1233:{\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}} }},} 2852:

found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem. This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.

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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.

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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.

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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.

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and it was too slow for scientists to practically apply it, this solution still left some issues unresolved. In the 1970s, implication to three-body from two-body forces had been discovered by

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for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes; therefore, it is currently necessary to approximate solutions by

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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

2952:, the motion of the Moon under the gravitational influence of Earth and the Sun. Later, this problem was also applied to other planets' interactions with the Earth and the Sun. 2422: 1079: 1050: 1662: 1264: 1904: 1858: 3108:

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

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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

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Mohr, R.F.; Furnstahl, R.J.; Hammer, H.-W.; Perry, R.J.; Wilson, K.G. (January 2006). "Precise numerical results for limit cycles in the quantum three-body problem".

1812: 5433: 1936: 1738:, 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 ( 285: 2911: 6237: 4783:

Hristov, Ivan; Hristova, Radoslava; Dmitrašinović, Veljko; Tanikawa, Kiyotaka (2024). "Three-body periodic collisionless equal-mass free-fall orbits revisited".

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Hristov, Ivan; Hristova, Radoslava; Dmitrašinović, Veljko; Tanikawa, Kiyotaka (2024). "Three-body periodic collisionless equal-mass free-fall orbits revisited".

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Krishnaswami, Govind S.; Senapati, Himalaya (2019). "An introduction to the classical three-body problem: From periodic solutions to instabilities and chaos".

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theory, in which the system is considered as a two-body problem plus additional forces causing deviations from a hypothetical unperturbed two-body trajectory.

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established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution.

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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

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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.

118: 2782:, with the heaviest body at the right angle and the lightest at the smaller acute angle. Burrau further investigated this problem in 1913. In 1967 144:. 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 2424:. However, this time dependence can be removed through a transformation to a rotating reference frame, which simplifies any subsequent analysis. 5947: 4023:

Li, Xiaoming; Liao, Shijun (December 2017). "More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits".

5707: 5361: 3432: 3136: 2733: 3278: 5551: 5234: 4837: 4583: 4364: 3450: 5662: 3568: 5900: 5277:

Chambers, J. E.; Wetherill, G. W. (1998). "Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions".

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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.

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Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities. The

5847: 1695: 31: 6180: 5181: 3856:

Chenciner, Alain; Montgomery, Richard (2000). "A remarkable periodic solution of the three-body problem in the case of equal masses".

1730: 6222: 5910: 3001:) 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. 186:

The mathematical statement of the three-body problem can be given in terms of the Newtonian equations of motion for vector positions

6415: 5209: 2845:

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

2557:, then not only can there be no triple collision, but the system is strictly bounded away from a triple collision. This implies, by 1717: 4544: 4751: 6097: 3101: 2274: 2572: 6539: 2517:

Using an appropriate change of variables to continue analyzing the solution beyond the binary collision, in a process known as

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in which Newton attempted to figure out if any long term stability is possible especially for such a system like that of our

2732:. These four families are the only known solutions for which there are explicit analytic formulae. In the special case of the 5481: 5152: 5026:"The vortex dynamics analogue of the restricted three-body problem: advection in the field of three identical point vortices" 3793: 3356: 3298: 3112:

and not the acceleration that is determined by their relative positions. As a consequence, the three-vortex problem is still

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he introduced later generations to the beginning of the gravitational three-body problem. In Proposition 66 of Book 1 of the

1699: 5915: 5905: 3123:. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the 2497:. In practice, the latter restriction is insignificant since initial conditions with zero angular momentum are rare, having 5431:

Li, Xiaoming; Liao, Shijun (2014). "On the stability of the three classes of Newtonian three-body planar periodic orbits".

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Beloriszky, D. (1930). "Application pratique des méthodes de M. Sundman à un cas particulier du problème des trois corps".

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found a zero angular momentum solution with three equal masses moving around a figure-eight shape. In 2000, mathematicians

1023:, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions 6529: 5394: 189: 6170: 6519: 6262: 6035: 5191: 4454:

Clairaut: "On the System of the World, according to the principles of Universal Gravitation" (at pp. 329–364); and

2885:) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as 6175: 6242: 6524: 5700: 4469: 3364: 2986: 2793: 114: 3466: 3392:

Barrow-Green, June (2008). "The Three-Body Problem". In Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.).

1688: 6290: 4286:"Analytical, Statistical Approximate Solution of Dissipative and Nondissipative Binary-Single Stellar Encounters" 42: 5994: 5939: 3250: 2518: 1755: 3165:. These problems have a global analytical solution in the form of a convergent power series, as was proven by 5392:

Hulkower, Neal D. (1980). "Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem".

6155: 5920: 5827: 5563: 5561:Šuvakov, Milovan; Dmitrašinović, V. (2013). "Three Classes of Newtonian Three-Body Planar Periodic Orbits". 3346: 2874: 58: 3482: 77: 6195: 5807: 5296: 2779: 2565:

for differential equations, that there are no complex singularities in a strip (depending on the value of

2454: 1012: 158:. Historically, the first specific three-body problem to receive extended study was the one involving the 129: 2787: 6514: 6345: 6252: 6050: 5877: 5812: 5787: 5693: 5324: 4548: 4127:

Li, Xiaoming; Liao, Shijun (2019). "Collisionless periodic orbits in the free-fall three-body problem".

3331: 3199: 3006: 2870: 2725: 2721: 2505: 1267: 1020: 1016: 132:, meaning there is no equation that always solves it. When three bodies orbit each other, the resulting 6107: 4599:

Efimov, V. (1970-12-21). "Energy levels arising from resonant two-body forces in a three-body system".

4086:"The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum" 3264: 3139:(i.e. for arbitrary ratio of masses) does not have a rigorous analytic solution in general relativity. 2369: 2905:

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when

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The circular restricted three-body problem is a valid approximation of elliptical orbits found in the

1055: 1026: 6355: 6185: 6007: 5852: 5641: 5582: 5496: 5452: 5403: 5370: 5333: 5288: 5265: 5090: 5037: 4990: 4947: 4908: 4873: 4802: 4608: 4503: 4309: 4199: 4146: 4042: 3966: 3875: 3808: 3684: 3645: 3614: 3499: 3438: 3203: 3132: 2983:

was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around Earth.

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found three families of periodic solutions in which the three masses are collinear at each instant.

2468: 3131:. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and 3069:

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the

1643: 1245: 6315: 6272: 6257: 6102: 6055: 6040: 6025: 5925: 5857: 5832: 5817: 5802: 5301: 4451:) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French): 4285: 4255: 3504: 3313: 3226: 3093: 3085: 3058:

In September 2023, several possible solutions have been found to the problem according to reports.

3013: 2933: 1863: 1817: 1767: 171: 98: 2728:

for the three-body problem. These solutions are valid for any mass ratios, and the masses move on

6493: 6360: 6077: 6072: 5964: 5842: 5744: 5606: 5572: 5468: 5442: 5419: 5255: 5122: 5080: 4792: 4732: 4712: 4570:. History of Mathematics. Vol. 11. Providence, Rhode Island: American Mathematical Society. 4527: 4493: 4325: 4299: 4263: 4189: 4162: 4136: 4097: 4066: 4032: 3956: 3899: 3891: 3865: 3757:(0) = (-0.93240737, -0.86473146). The values are obtained from Chenciner & Montgomery (2000). 3536: 3351: 3270: 3195: 3120: 2976: 2886: 2878: 2826: 2366:. In this form the equations of motion carry an explicit time dependence through the coordinates 5025: 5675: 4225: 2797:

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

6380: 6365: 6330: 6320: 6217: 5837: 5759: 5598: 5547: 5512: 5230: 5114: 5106: 5053: 5006: 4981:

Aref, Hassan; Pomphrey, Neil (1980-08-18). "Integrable and chaotic motions of four vortices".

4963: 4843: 4833: 4701:"Newton versus the machine: Solving the chaotic three-body problem using deep neural networks" 4681: 4663: 4624: 4579: 4519: 4484: 4370: 4360: 4058: 3824: 3446: 3323: 3318: 3230: 3207: 3113: 2772: 2562: 2472: 1759: 175: 141: 4864:

Crandall, R.; Whitnell, R.; Bettega, R. (1984). "Exactly soluble two-electron atomic model".

2691:{\displaystyle \sigma ={\frac {e^{\frac {\pi s}{2\beta }}-1}{e^{\frac {\pi s}{2\beta }}+1}}.} 1784: 6473: 6385: 6335: 6232: 6160: 6112: 5989: 5974: 5969: 5764: 5649: 5590: 5504: 5460: 5411: 5378: 5341: 5306: 5279: 5098: 5045: 4998: 4955: 4916: 4881: 4810: 4722: 4671: 4655: 4616: 4571: 4511: 4395: 4317: 4207: 4154: 4107: 4050: 3974: 3883: 3816: 3692: 3653: 3546: 3341: 3336: 3293: 3021: 2956: 2838:

event resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.

2814: 2783: 2737: 2498: 2494: 1758:, the two co-orbiting bodies are stationary, and the third can be stationary as well at the 145: 133: 125: 3213:). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum 3016:

worked on the restricted problem in the late 19th century with an application of motion of

1909: 263: 6534: 6405: 6300: 6227: 6060: 5872: 5862: 5679: 5246: 4699:

Breen, Philip G.; Foley, Christopher N.; Boekholt, Tjarda; Portegies Zwart, Simon (2020).

4290: 3166: 3028: 2994: 2990: 2960: 2941: 2830: 1763: 1743: 1739: 6395: 6340: 2810: 2578:

Find a conformal transformation that maps this strip into the unit disc. For example, if

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at an instant) and triple collisions (collisions between three particles at an instant).

5645: 5586: 5500: 5456: 5407: 5374: 5337: 5292: 5269: 5102: 5094: 5041: 4994: 4951: 4912: 4877: 4806: 4643: 4612: 4507: 4313: 4203: 4150: 4046: 3970: 3918: 3879: 3812: 3688: 3649: 3618: 3442: 6488: 6455: 6450: 6445: 6247: 6137: 6132: 6030: 5979: 5895: 5769: 5540: 5143: 4676: 3308: 3303: 3284: 3188: 3151: 3088:. Like the gravitational three-body problem, the helium atom cannot be solved exactly. 3082: 3052: 2882: 2736:, these solutions, viewed in a frame rotating with the primaries, become points called 2714: 2476: 1941: 1771: 994: 152: 82: 4419: 2801: 6508: 6483: 6440: 6430: 6425: 6325: 6305: 6165: 6087: 5984: 5792: 5610: 5423: 5244:

Bagla, J. S. (2005). "Cosmological N-body simulation: Techniques, scope and status".

5049: 5002: 4736: 4620: 4531: 4329: 3979: 3944: 3407: 3124: 3037: 3033: 2972: 2446: 1774:, and the planetoid is assumed to move in the plane defined by the circular orbits. 5472: 5126: 4166: 4070: 3903: 6435: 6400: 6310: 6267: 6122: 6117: 5716: 5632: 5594: 5139: 4756: 3177: 2980: 2964: 2949: 2906: 2890: 2729: 2480: 2442: 1735: 137: 6082: 5520: 3832: 3592: 5682:– an example of a computer program that solves the three-body problem numerically 4935: 4158: 1770:. Usually this two-body motion is taken to consist of circular orbits around the 113:

that orbit each other in space and calculate their subsequent trajectories using

6420: 6410: 6295: 6045: 5867: 5774: 5508: 3820: 3070: 3044: 3009: 2937: 2929: 2894: 2869:

Using a computer, the problem may be solved to arbitrarily high precision using

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domain of stability is small. For instance, the probability of a binary–binary

6478: 6375: 5822: 5654: 5627: 5464: 4515: 4374: 4054: 3550: 3260: 3242: 3128: 2835: 2822: 110: 17: 5110: 5057: 5010: 4967: 4667: 4628: 4523: 4399: 4062: 3657: 6390: 6350: 6092: 5754: 5739: 4847: 4727: 4700: 3995: 3078: 5602: 5516: 5310: 5118: 4685: 4562: 4389: 3828: 4899:

Calogero, F. (1969). "Solution of a Three-Body Problem in One Dimension".

4112: 4085: 3996:"New periodic solutions to the three-body problem and gravitational waves" 2437: 6127: 5260: 4659: 3961: 3055:

solver for the three-body problem, trained using a numerical integrator.

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singularities from the transformed equations for the three-body problem.

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Master of Science Thesis at the Faculty of Physics, Belgrade University

3895: 3569:"Physicists Discover a Whopping 13 New Solutions to Three-Body Problem" 3238: 3162: 2462: 1702: in this section. Unsourced material may be challenged and removed. 94: 4920: 1664:

is simply the total energy of the system, gravitational plus kinetic.

170:. In an extended modern sense, a three-body problem is any problem in 6370: 5663:

Physicists Discover a Whopping 13 New Solutions to Three-Body Problem

5546:(3 vol. English translated ed.). American Institute of Physics. 4959: 4752:"We Just Got 12,000 New Solutions to The Infamous Three-Body Problem" 3870: 3222: 3105: 3074: 2571:) in the complex plane centered around the real axis (related to the 2558: 4885: 3887: 3633: 2524:

Proving that triple collisions only occur when the angular momentum

5346: 5319: 5085: 4797: 4717: 4304: 4194: 4141: 4102: 4037: 3697: 3672: 3541: 1814:

be the masses of the two massive bodies, with (planar) coordinates

5577: 5447: 3634:"Numerische Berechnung eines Spezialfalles des Dreikörperproblems" 3234: 3017: 2917: 2877:

time. There have been attempts of creating computer programs that

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Efstathiou, G.; Davis, M.; White, S. D. M.; Frenk, C. S. (1985).

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The gravitational three-body problem has also been studied using

5320:"Numerical techniques for large cosmological N-body simulations" 2921: 163: 5689: 5685: 5071:

Musielak, Z. E.; Quarles, B. (2014). "The three-body problem".

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An important issue in proving this result is the fact that the

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The 1747 memoirs of both parties can be read in the volume of

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This article is about the physics theory. For other uses, see

3249:-body systems. Some applications are conveniently treated by 1649: 1281: 1251: 1202: 1130: 4408:– via Hong Kong University of Science and Technology. 2805:

20 examples of periodic solutions to the three-body problem

2359:{\displaystyle r_{i}={\sqrt {(x-x_{i})^{2}+(y-y_{i})^{2}}}} 2441:

While a system of 3 bodies interacting gravitationally is

4545:"Coplanar Motion of Two Planets, One Having a Zero Mass" 4084:

Li, Xiaoming; Jing, Yipeng; Liao, Shijun (August 2018).

3673:"Complete Solution of a General Problem of Three Bodies" 260:

of three gravitationally interacting bodies with masses

4284:

Ginat, Yonadav Barry; Perets, Hagai B. (23 July 2021).

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objects move under one of the physical forces, such as

2493:, except for initial conditions corresponding to zero 1781:

Mathematically, the problem is stated as follows. Let

1778:

the historical development of the three-body problem.

4644:"On the periodic solutions of the three-body problem" 3217:-body problem. Among classical physical systems, the 2608: 2372: 2277: 1966: 1944: 1912: 1866: 1820: 1787: 1646: 1278: 1248: 1089: 1058: 1029: 1019:. The problem can also be stated equivalently in the 997: 295: 266: 192: 27:

Physics problem related to laws of motion and gravity

4233: 3387: 3385: 3383: 253:{\displaystyle \mathbf {r_{i}} =(x_{i},y_{i},z_{i})} 105:

is to take the initial positions and velocities (or

6464: 6281: 6208: 6146: 6016: 6003: 5955: 5886: 5730: 5723: 5672: 2873:although high precision requires a large amount of 2588:(the new variable after the regularization) and if 5539: 3715:has been set to 1, and the initial conditions are 3426: 3424: 3005:problem, providing scientists a rare example of a 2768:being symmetric instances of Lagrange's solution. 2690: 2416: 2358: 2261: 1958:. Then, the motion of the planetoid is given by: 1950: 1930: 1898: 1852: 1806: 1656: 1630: 1258: 1232: 1073: 1044: 1003: 983: 279: 252: 4705:Monthly Notices of the Royal Astronomical Society 4357:The Three-body Problem from Pythagoras to Hawking 4090:Publications of the Astronomical Society of Japan 3949:Monthly Notices of the Royal Astronomical Society 5434:Science China Physics, Mechanics & Astronomy 4025:Science China Physics, Mechanics & Astronomy 3437:. American Mathematical Society. pp. 8–12. 3150:The three-body problem is a special case of the 151:The three-body problem is a special case of the 4859: 4857: 3396:. Princeton University Press. pp. 726–728. 5030:Journal of Physics A: Mathematical and General 3671:Victor Szebehely; C. Frederick Peters (1967). 2881:the three-body problem (and by extension, the 2705:computations would involve at least 10 terms. 2701:This finishes the proof of Sundman's theorem. 5701: 3562: 3560: 2530:vanishes. By restricting the initial data to 8: 3926:Notices of the American Mathematical Society 2955:The physical problem was first addressed by 4830:Introduction to Quantum Mechanics (2nd ed.) 4785:Celestial Mechanics and Dynamical Astronomy 4391:Philosophiæ naturalis principia mathematica 4182:Celestial Mechanics and Dynamical Astronomy 3945:"A new outcome of binary–binary scattering" 2912:Philosophiæ Naturalis Principia Mathematica 2475:to the three-body problem in the form of a 178:that models the motion of three particles. 57:. Relevant discussion may be found on the 6013: 5727: 5708: 5694: 5686: 3919:"A new solution to the three-body problem" 5653: 5576: 5446: 5382: 5345: 5300: 5259: 5210:Learn how and when to remove this message 5084: 4796: 4726: 4716: 4675: 4642:Liao, Shijun; Li, Xiaoming (2019-11-01). 4497: 4303: 4256:"A Centuries-Old Physics Mystery? Solved" 4193: 4140: 4111: 4101: 4036: 3978: 3960: 3869: 3696: 3540: 2655: 2622: 2615: 2607: 2399: 2377: 2371: 2348: 2338: 2316: 2306: 2291: 2282: 2276: 2244: 2239: 2228: 2215: 2209: 2194: 2189: 2178: 2165: 2159: 2140: 2122: 2115: 2100: 2095: 2084: 2071: 2065: 2050: 2045: 2034: 2021: 2015: 1996: 1978: 1971: 1967: 1965: 1943: 1911: 1887: 1874: 1865: 1841: 1828: 1819: 1792: 1786: 1762:, or move around them, for instance on a 1718:Learn how and when to remove this message 1648: 1647: 1645: 1616: 1602: 1595: 1590: 1587: 1575: 1561: 1554: 1549: 1546: 1534: 1520: 1513: 1508: 1505: 1494: 1487: 1482: 1472: 1467: 1462: 1454: 1444: 1434: 1423: 1416: 1411: 1401: 1396: 1391: 1383: 1373: 1363: 1352: 1345: 1340: 1330: 1325: 1320: 1312: 1302: 1292: 1280: 1279: 1277: 1250: 1249: 1247: 1217: 1212: 1201: 1200: 1194: 1170: 1165: 1159: 1145: 1140: 1129: 1128: 1122: 1101: 1096: 1090: 1088: 1064: 1059: 1057: 1035: 1030: 1028: 996: 965: 960: 952: 947: 937: 932: 927: 918: 913: 903: 898: 895: 889: 870: 865: 857: 852: 842: 837: 832: 823: 818: 808: 803: 800: 794: 770: 769: 758: 756: 755: 738: 733: 725: 720: 710: 705: 700: 691: 686: 676: 671: 668: 662: 643: 638: 630: 625: 615: 610: 605: 596: 591: 581: 576: 573: 567: 543: 542: 531: 529: 528: 511: 506: 498: 493: 483: 478: 473: 464: 459: 449: 444: 441: 435: 416: 411: 403: 398: 388: 383: 378: 369: 364: 354: 349: 346: 340: 316: 315: 304: 302: 301: 296: 294: 271: 265: 241: 228: 215: 198: 193: 191: 5229:. New York: Cambridge University Press. 2800: 2792: 2436: 1729: 128:, the three-body problem has no general 76: 3595:, Historia Mathematica 37, pp. 164–203. 3379: 3051:In 2019, Breen et al. announced a fast 3027:At the beginning of the 20th century, 2897:of different outcomes may be computed. 5362:Indiana University Mathematics Journal 4764:from the original on 24 September 2023 3765: 3763: 3408:"Historical Notes: Three-Body Problem" 3394:The Princeton Companion to Mathematics 3012:at the beginning of the 21st century. 2734:circular restricted three-body problem 4350: 4348: 4346: 3062:Other problems involving three bodies 2848:In 2017, researchers Xiaoming Li and 2489:. This series converges for all real 1015:. This is a set of nine second-order 119:Newton's law of universal gravitation 7: 5357:"The Zero Energy Three Body Problem" 4750:Watson, Claire (23 September 2023). 3750:(0) = (0.4662036850, 0.4323657300); 1700:adding citations to reliable sources 5848:Measure-preserving dynamical system 4564:Poincaré and the Three Body Problem 3434:Poincaré and the Three Body Problem 2889:. In addition, using the theory of 2445:, a system of 3 bodies interacting 1766:. It can be useful to consider the 32:Three-body problem (disambiguation) 5542:New Methods of Celestial Mechanics 3221:-body problem usually refers to a 3133:sophisticated numerical techniques 1209: 1197: 1137: 1125: 25: 6416:Oleksandr Mykolayovych Sharkovsky 5024:Neufeld, Z; Tél, T (1997-03-21). 4561:Barrow-Green, June (1996-10-29). 3729:(0) = (-0.97000436, 0.24308753); 3487:Eric Weisstein's World of Physics 2969:determination of longitude at sea 2771:In work summarized in 1892–1899, 2417:{\displaystyle x_{i}(t),y_{i}(t)} 5946: 5938: 5227:Gravitational n-Body Simulations 5171: 3980:10.1046/j.1365-8711.2000.04027.x 3711:Here the gravitational constant 3567:Cartwright, Jon (8 March 2013). 3277: 3263: 1742:are dynamic and not shown). The 1676: 1596: 1592: 1555: 1551: 1514: 1510: 1488: 1484: 1473: 1469: 1417: 1413: 1402: 1398: 1346: 1342: 1331: 1327: 1218: 1214: 1171: 1167: 1146: 1142: 1102: 1098: 1074:{\displaystyle \mathbf {p_{i}} } 1065: 1061: 1045:{\displaystyle \mathbf {r_{i}} } 1036: 1032: 953: 949: 938: 934: 919: 915: 904: 900: 858: 854: 843: 839: 824: 820: 809: 805: 771: 759: 726: 722: 711: 707: 692: 688: 677: 673: 631: 627: 616: 612: 597: 593: 582: 578: 544: 532: 499: 495: 484: 480: 465: 461: 450: 446: 404: 400: 389: 385: 370: 366: 355: 351: 317: 305: 199: 195: 41: 4901:Journal of Mathematical Physics 3770:Šuvakov, M.; Dmitrašinović, V. 3593:The dramatic episode of Sundman 3483:"Restricted Three-Body Problem" 1687:needs additional citations for 1158: 87:law of conservation of momentum 6181:Rabinovich–Fabrikant equations 5595:10.1103/PhysRevLett.110.114301 5482:"Braids in Classical Dynamics" 5153:The Mathematical Intelligencer 5073:Reports on Progress in Physics 4832:. Prentice Hall. p. 311. 3794:"Braids in classical dynamics" 3299:Galaxy formation and evolution 3202:or, for some cases, classical 2979:. However the accuracy of the 2411: 2405: 2389: 2383: 2345: 2325: 2313: 2293: 1925: 1913: 1893: 1867: 1847: 1821: 1657:{\displaystyle {\mathcal {H}}} 1495: 1463: 1424: 1392: 1353: 1321: 1259:{\displaystyle {\mathcal {H}}} 961: 928: 866: 833: 734: 701: 639: 606: 507: 474: 412: 379: 247: 208: 55:of this article is in question 1: 5194:and help improve the section. 5103:10.1088/0034-4885/77/6/065901 4394:. London: G. & J. Innys. 3994:Hudomal, Ana (October 2015). 1899:{\displaystyle (x_{2},y_{2})} 1853:{\displaystyle (x_{1},y_{1})} 1752:restricted three-body problem 1668:Restricted three-body problem 5003:10.1016/0375-9601(80)90375-8 4828:Griffiths, David J. (2004). 4621:10.1016/0370-2693(70)90349-7 4551:, Vol. III, pp. 65–73, 1887. 4159:10.1016/j.newast.2019.01.003 3917:Montgomery, Richard (2001). 3135:are required. Even the full 2602:, then this map is given by 2471:proved that there exists an 5916:Poincaré recurrence theorem 5509:10.1103/PhysRevLett.70.3675 4934:Aref, Hassan (1979-03-01). 4866:American Journal of Physics 4254:Technion (6 October 2021). 3943:Heggie, Douglas C. (2000). 3821:10.1103/PhysRevLett.70.3675 3431:Barrow-Green, June (1997). 2573:Cauchy–Kovalevskaya theorem 6556: 5911:Poincaré–Bendixson theorem 5480:Moore, Cristopher (1993). 5384:10.1512/iumj.1978.27.27030 5355:Hulkower, Neal D. (1978). 5050:10.1088/0305-4470/30/6/043 4936:"Motion of three vortices" 4815:10.1007/s10569-023-10177-w 4322:10.1103/PhysRevX.11.031020 4212:10.1007/s10569-023-10177-w 3792:Moore, Cristopher (1993). 3081:interact according to the 29: 6263:Swinging Atwood's machine 5936: 5906:Krylov–Bogolyubov theorem 5783: 5655:10.4249/scholarpedia.2111 5626:Chenciner, Alain (2007). 5465:10.1007/s11433-014-5563-5 4516:10.1016/j.aop.2005.10.002 4055:10.1007/s11433-017-9078-5 3638:Astronomische Nachrichten 3591:Barrow-Green, J. (2010). 3551:10.1007/s12045-019-0760-1 3245:, can also be treated as 85:, in accordance with the 6171:Lotka–Volterra equations 5995:Synchronization of chaos 5798:axiom A dynamical system 4355:Valtonen, Mauri (2016). 3658:10.1002/asna.19131950602 3500:"The Three-Body Problem" 3467:"The Three-Body Problem" 2999:Problème des trois Corps 2971:, solved in practice by 1756:rotating reference frame 182:Mathematical description 6156:Double scroll attractor 5921:Stable manifold theorem 5828:False nearest neighbors 5564:Physical Review Letters 5489:Physical Review Letters 5225:Aarseth, S. J. (2003). 4648:National Science Review 4470:Jean le Rond d'Alembert 3801:Physical Review Letters 3347:Synodic reference frame 2987:Jean le Rond d'Alembert 1807:{\displaystyle m_{1,2}} 115:Newton's laws of motion 6540:Equations of astronomy 6196:Van der Pol oscillator 6176:Mackey–Glass equations 5808:Box-counting dimension 5311:10.1006/icar.1998.6007 4400:10.14711/spcol/b706487 4388:Newton, Isaac (1726). 3358:The Three-Body Problem 3157:, which describes how 3036:, which was named the 2998: 2806: 2798: 2726:central configurations 2709:Special-case solutions 2692: 2483:in terms of powers of 2450: 2418: 2360: 2263: 1952: 1932: 1900: 1854: 1808: 1747: 1658: 1632: 1260: 1234: 1075: 1046: 1017:differential equations 1013:gravitational constant 1005: 985: 281: 254: 90: 6346:Svetlana Jitomirskaya 6253:Multiscroll attractor 6098:Interval exchange map 6051:Dyadic transformation 6036:Complex quadratic map 5878:Topological conjugacy 5813:Correlation dimension 5788:Anosov diffeomorphism 5538:Poincaré, H. (1967). 5325:Astrophysical Journal 5144:"The Solution of the 4940:The Physics of Fluids 4728:10.1093/mnras/staa713 4549:Annals of Mathematics 3858:Annals of Mathematics 3607:Bulletin Astronomique 3332:Symplectic integrator 3200:numerical integration 3007:renormalization group 2871:numerical integration 2804: 2796: 2693: 2513:the following steps: 2506:radius of convergence 2461:However, in 1912 the 2440: 2419: 2361: 2264: 1953: 1933: 1931:{\displaystyle (x,y)} 1901: 1855: 1809: 1733: 1659: 1633: 1261: 1235: 1076: 1047: 1021:Hamiltonian formalism 1006: 986: 282: 280:{\displaystyle m_{i}} 255: 80: 6530:Mathematical physics 6356:Edward Norton Lorenz 5628:"Three body problem" 3772:"Three-body Gallery" 3677:Astronomical Journal 3535:. Springer: 87–114. 3206:approximations (see 3204:trigonometric series 3167:Karl F. Sundman 2975:'s invention of the 2959:and subsequently by 2865:Numerical approaches 2780:3:4:5 right triangle 2606: 2469:Karl Fritiof Sundman 2455:closed-form solution 2453:There is no general 2370: 2275: 1964: 1942: 1910: 1864: 1818: 1785: 1696:improve this article 1644: 1276: 1246: 1087: 1056: 1027: 995: 293: 264: 190: 130:closed-form solution 6520:Classical mechanics 6316:Mitchell Feigenbaum 6258:Population dynamics 6243:Hénon–Heiles system 6103:Irrational rotation 6056:Dynamical billiards 6041:Coupled map lattice 5901:Liouville's theorem 5833:Hausdorff dimension 5818:Conservative system 5803:Bifurcation diagram 5646:2007SchpJ...2.2111C 5587:2013PhRvL.110k4301S 5501:1993PhRvL..70.3675M 5457:2014SCPMA..57.2121L 5408:1980CeMec..21...37H 5395:Celestial Mechanics 5375:1978IUMJ...27..409H 5338:1985ApJS...57..241E 5293:1998Icar..136..304C 5270:2005CSci...88.1088B 5095:2014RPPh...77f5901M 5042:1997JPhA...30.2263N 4995:1980PhLA...78..297A 4952:1979PhFl...22..393A 4913:1969JMP....10.2191C 4878:1984AmJPh..52..438C 4807:2024CeMDA.136....7H 4613:1970PhLB...33..563E 4508:2006AnPhy.321..225M 4314:2021PhRvX..11c1020G 4204:2024CeMDA.136....7H 4151:2019NewA...70...22L 4113:10.1093/pasj/psy057 4047:2017SCPMA..60l9511L 3971:2000MNRAS.318L..61H 3880:2000math.....11268C 3813:1993PhRvL..70.3675M 3689:1967AJ.....72..876S 3650:1913AN....195..113B 3619:1930BuAst...6..417B 3505:Scientific American 3489:. Wolfram Research. 3443:1997ptbp.book.....B 3314:Low-energy transfer 3227:cluster of galaxies 3094:harmonic attraction 3086:Coulomb interaction 3014:George William Hill 2934:Nicolaus Copernicus 2821:In 1993, physicist 2788:C. Frederick Peters 2249: 2199: 2105: 2055: 1768:effective potential 172:classical mechanics 99:classical mechanics 89:, remains in place. 6494:Santa Fe Institute 6361:Aleksandr Lyapunov 6191:Three-body problem 6078:Gingerbreadman map 5965:Bifurcation theory 5843:Lyapunov stability 5678:2022-11-17 at the 5416:10.1007/BF01230244 4660:10.1093/nsr/nwz102 4420:"Amerigo Vespucci" 3352:Triple star system 3271:Mathematics portal 3196:numerical analysis 3121:general relativity 3102:point vortex model 2977:marine chronometer 2887:special relativity 2827:Santa Fe Institute 2807: 2799: 2730:Keplerian ellipses 2688: 2451: 2414: 2356: 2259: 2257: 2235: 2185: 2091: 2041: 1948: 1928: 1896: 1850: 1804: 1748: 1654: 1628: 1256: 1230: 1071: 1042: 1001: 981: 979: 277: 250: 142:initial conditions 103:three-body problem 91: 6525:Dynamical systems 6502: 6501: 6366:Benoît Mandelbrot 6331:Martin Gutzwiller 6321:Peter Grassberger 6204: 6203: 6186:Rössler attractor 5934: 5933: 5838:Invariant measure 5760:Lyapunov exponent 5553:978-1-56396-117-5 5495:(24): 3675–3679. 5441:(11): 2121–2126. 5236:978-0-521-43272-6 5220: 5219: 5212: 4983:Physics Letters A 4921:10.1063/1.1664820 4907:(12): 2191–2196. 4839:978-0-13-111892-8 4601:Physics Letters B 4585:978-0-8218-0367-7 4576:10.1090/hmath/011 4485:Annals of Physics 4366:978-3-319-22726-9 4226:"3body simulator" 3860:. Second Series. 3807:(24): 3675–3679. 3452:978-0-8218-0367-7 3319:Michael Minovitch 3231:planetary systems 2893:, an approximate 2879:numerically solve 2738:Lagrangian points 2683: 2673: 2640: 2563:existence theorem 2540:, he removed all 2479:, specifically a 2473:analytic solution 2354: 2250: 2200: 2147: 2106: 2056: 2003: 1951:{\displaystyle 1} 1760:Lagrangian points 1728: 1727: 1720: 1623: 1582: 1541: 1500: 1429: 1358: 1225: 1186: 1153: 1117: 1004:{\displaystyle G} 972: 877: 766: 745: 650: 539: 518: 423: 312: 176:quantum mechanics 146:numerical methods 75: 74: 16:(Redirected from 6547: 6474:Butterfly effect 6386:Itamar Procaccia 6336:Brosl Hasslacher 6233:Elastic pendulum 6161:Duffing equation 6108:Kaplan–Yorke map 6026:Arnold's cat map 6014: 5990:Stability theory 5975:Dynamical system 5970:Control of chaos 5950: 5942: 5926:Takens's theorem 5858:Poincaré section 5728: 5710: 5703: 5696: 5687: 5659: 5657: 5614: 5580: 5557: 5545: 5534: 5532: 5531: 5525: 5519:. Archived from 5486: 5476: 5450: 5427: 5388: 5386: 5351: 5349: 5314: 5304: 5273: 5263: 5261:astro-ph/0411043 5240: 5215: 5208: 5204: 5201: 5195: 5190:Please read the 5186:may need cleanup 5175: 5174: 5167: 5157: 5137: 5131: 5130: 5088: 5068: 5062: 5061: 5036:(6): 2263–2280. 5021: 5015: 5014: 4978: 4972: 4971: 4960:10.1063/1.862605 4931: 4925: 4924: 4896: 4890: 4889: 4861: 4852: 4851: 4825: 4819: 4818: 4800: 4780: 4774: 4773: 4771: 4769: 4747: 4741: 4740: 4730: 4720: 4711:(2): 2465–2470. 4696: 4690: 4689: 4679: 4654:(6): 1070–1071. 4639: 4633: 4632: 4596: 4590: 4589: 4569: 4558: 4552: 4542: 4536: 4535: 4501: 4479: 4473: 4467: 4461: 4441: 4435: 4434: 4432: 4431: 4416: 4410: 4409: 4407: 4406: 4385: 4379: 4378: 4352: 4341: 4340: 4338: 4336: 4307: 4281: 4275: 4274: 4272: 4270: 4251: 4245: 4244: 4242: 4241: 4232:. Archived from 4222: 4216: 4215: 4197: 4177: 4171: 4170: 4144: 4124: 4118: 4117: 4115: 4105: 4081: 4075: 4074: 4040: 4020: 4014: 4013: 4011: 4009: 4000: 3991: 3985: 3984: 3982: 3964: 3962:astro-ph/9604016 3940: 3934: 3933: 3923: 3914: 3908: 3907: 3873: 3853: 3847: 3846: 3844: 3843: 3837: 3831:. Archived from 3798: 3789: 3783: 3782: 3780: 3778: 3767: 3758: 3709: 3703: 3702: 3700: 3668: 3662: 3661: 3629: 3623: 3622: 3602: 3596: 3589: 3583: 3582: 3580: 3579: 3564: 3555: 3554: 3544: 3524: 3518: 3517: 3515: 3513: 3496: 3490: 3480: 3474: 3473: 3471: 3463: 3457: 3456: 3428: 3419: 3418: 3416: 3414: 3404: 3398: 3397: 3389: 3342:Two-body problem 3337:Sitnikov problem 3327:-body simulation 3326: 3294:Few-body systems 3287: 3282: 3281: 3273: 3268: 3267: 3248: 3220: 3216: 3211:-body simulation 3210: 3191: 3186: 3175: 3160: 3154: 3145: 3137:two-body problem 3104:, the motion of 3077:nucleus and two 2957:Amerigo Vespucci 2928:Guided by major 2815:Roger A. Broucke 2784:Victor Szebehely 2697: 2695: 2694: 2689: 2684: 2682: 2675: 2674: 2672: 2664: 2656: 2649: 2642: 2641: 2639: 2631: 2623: 2616: 2601: 2596: 2587: 2570: 2556: 2547:Showing that if 2539: 2529: 2499:Lebesgue measure 2495:angular momentum 2492: 2488: 2433:General solution 2423: 2421: 2420: 2415: 2404: 2403: 2382: 2381: 2365: 2363: 2362: 2357: 2355: 2353: 2352: 2343: 2342: 2321: 2320: 2311: 2310: 2292: 2287: 2286: 2268: 2266: 2265: 2260: 2258: 2251: 2248: 2243: 2234: 2233: 2232: 2216: 2214: 2213: 2201: 2198: 2193: 2184: 2183: 2182: 2166: 2164: 2163: 2148: 2146: 2145: 2144: 2131: 2127: 2126: 2116: 2107: 2104: 2099: 2090: 2089: 2088: 2072: 2070: 2069: 2057: 2054: 2049: 2040: 2039: 2038: 2022: 2020: 2019: 2004: 2002: 2001: 2000: 1987: 1983: 1982: 1972: 1957: 1955: 1954: 1949: 1937: 1935: 1934: 1929: 1905: 1903: 1902: 1897: 1892: 1891: 1879: 1878: 1859: 1857: 1856: 1851: 1846: 1845: 1833: 1832: 1813: 1811: 1810: 1805: 1803: 1802: 1740:Coriolis effects 1723: 1716: 1712: 1709: 1703: 1680: 1672: 1663: 1661: 1660: 1655: 1653: 1652: 1637: 1635: 1634: 1629: 1624: 1622: 1621: 1620: 1607: 1606: 1601: 1600: 1599: 1588: 1583: 1581: 1580: 1579: 1566: 1565: 1560: 1559: 1558: 1547: 1542: 1540: 1539: 1538: 1525: 1524: 1519: 1518: 1517: 1506: 1501: 1499: 1498: 1493: 1492: 1491: 1478: 1477: 1476: 1466: 1460: 1459: 1458: 1449: 1448: 1435: 1430: 1428: 1427: 1422: 1421: 1420: 1407: 1406: 1405: 1395: 1389: 1388: 1387: 1378: 1377: 1364: 1359: 1357: 1356: 1351: 1350: 1349: 1336: 1335: 1334: 1324: 1318: 1317: 1316: 1307: 1306: 1293: 1285: 1284: 1265: 1263: 1262: 1257: 1255: 1254: 1239: 1237: 1236: 1231: 1226: 1224: 1223: 1222: 1221: 1207: 1206: 1205: 1195: 1187: 1185: 1177: 1176: 1175: 1174: 1160: 1154: 1152: 1151: 1150: 1149: 1135: 1134: 1133: 1123: 1118: 1116: 1108: 1107: 1106: 1105: 1091: 1080: 1078: 1077: 1072: 1070: 1069: 1068: 1051: 1049: 1048: 1043: 1041: 1040: 1039: 1010: 1008: 1007: 1002: 990: 988: 987: 982: 980: 973: 971: 970: 969: 964: 958: 957: 956: 943: 942: 941: 931: 925: 924: 923: 922: 909: 908: 907: 896: 894: 893: 878: 876: 875: 874: 869: 863: 862: 861: 848: 847: 846: 836: 830: 829: 828: 827: 814: 813: 812: 801: 799: 798: 776: 775: 774: 768: 767: 762: 757: 746: 744: 743: 742: 737: 731: 730: 729: 716: 715: 714: 704: 698: 697: 696: 695: 682: 681: 680: 669: 667: 666: 651: 649: 648: 647: 642: 636: 635: 634: 621: 620: 619: 609: 603: 602: 601: 600: 587: 586: 585: 574: 572: 571: 549: 548: 547: 541: 540: 535: 530: 519: 517: 516: 515: 510: 504: 503: 502: 489: 488: 487: 477: 471: 470: 469: 468: 455: 454: 453: 442: 440: 439: 424: 422: 421: 420: 415: 409: 408: 407: 394: 393: 392: 382: 376: 375: 374: 373: 360: 359: 358: 347: 345: 344: 322: 321: 320: 314: 313: 308: 303: 286: 284: 283: 278: 276: 275: 259: 257: 256: 251: 246: 245: 233: 232: 220: 219: 204: 203: 202: 155: 134:dynamical system 126:two-body problem 70: 67: 45: 37: 21: 6555: 6554: 6550: 6549: 6548: 6546: 6545: 6544: 6505: 6504: 6503: 6498: 6466: 6460: 6406:Caroline Series 6301:Mary Cartwright 6283: 6277: 6228:Double pendulum 6210: 6200: 6149: 6142: 6068:Exponential map 6019: 6005: 5999: 5957: 5951: 5944: 5930: 5896:Ergodic theorem 5889: 5882: 5873:Stable manifold 5863:Recurrence plot 5779: 5733: 5719: 5714: 5680:Wayback Machine 5673:3body simulator 5625: 5622: 5617: 5560: 5554: 5537: 5529: 5527: 5523: 5484: 5479: 5430: 5391: 5354: 5317: 5276: 5247:Current Science 5243: 5237: 5224: 5216: 5205: 5199: 5196: 5189: 5182:Further reading 5176: 5172: 5165: 5163:Further reading 5160: 5138: 5134: 5070: 5069: 5065: 5023: 5022: 5018: 4980: 4979: 4975: 4933: 4932: 4928: 4898: 4897: 4893: 4886:10.1119/1.13650 4863: 4862: 4855: 4840: 4827: 4826: 4822: 4782: 4781: 4777: 4767: 4765: 4749: 4748: 4744: 4698: 4697: 4693: 4641: 4640: 4636: 4598: 4597: 4593: 4586: 4567: 4560: 4559: 4555: 4543: 4539: 4499:nucl-th/0509076 4481: 4480: 4476: 4468: 4464: 4442: 4438: 4429: 4427: 4418: 4417: 4413: 4404: 4402: 4387: 4386: 4382: 4367: 4354: 4353: 4344: 4334: 4332: 4291:Physical Review 4283: 4282: 4278: 4268: 4266: 4253: 4252: 4248: 4239: 4237: 4230:3body simulator 4224: 4223: 4219: 4179: 4178: 4174: 4126: 4125: 4121: 4083: 4082: 4078: 4022: 4021: 4017: 4007: 4005: 3998: 3993: 3992: 3988: 3942: 3941: 3937: 3921: 3916: 3915: 3911: 3888:10.2307/2661357 3855: 3854: 3850: 3841: 3839: 3835: 3796: 3791: 3790: 3786: 3776: 3774: 3769: 3768: 3761: 3756: 3749: 3742: 3735: 3728: 3721: 3710: 3706: 3670: 3669: 3665: 3632:Burrau (1913). 3631: 3630: 3626: 3604: 3603: 3599: 3590: 3586: 3577: 3575: 3566: 3565: 3558: 3526: 3525: 3521: 3511: 3509: 3498: 3497: 3493: 3481: 3477: 3469: 3465: 3464: 3460: 3453: 3430: 3429: 3422: 3412: 3410: 3406: 3405: 3401: 3391: 3390: 3381: 3377: 3372: 3324: 3283: 3276: 3269: 3262: 3259: 3246: 3218: 3214: 3208: 3198:in the form of 3189: 3181: 3170: 3158: 3152: 3148: 3143: 3064: 2991:Alexis Clairaut 2961:Galileo Galilei 2942:Johannes Kepler 2903: 2867: 2831:Alain Chenciner 2767: 2763: 2759: 2755: 2751: 2747: 2743: 2711: 2665: 2657: 2651: 2650: 2632: 2624: 2618: 2617: 2604: 2603: 2591: 2589: 2579: 2566: 2548: 2531: 2525: 2490: 2484: 2435: 2430: 2395: 2373: 2368: 2367: 2344: 2334: 2312: 2302: 2278: 2273: 2272: 2256: 2255: 2224: 2217: 2205: 2174: 2167: 2155: 2136: 2132: 2118: 2117: 2112: 2111: 2080: 2073: 2061: 2030: 2023: 2011: 1992: 1988: 1974: 1973: 1962: 1961: 1940: 1939: 1908: 1907: 1883: 1870: 1862: 1861: 1837: 1824: 1816: 1815: 1788: 1783: 1782: 1764:horseshoe orbit 1744:Lagrange points 1724: 1713: 1707: 1704: 1693: 1681: 1670: 1642: 1641: 1612: 1608: 1591: 1589: 1571: 1567: 1550: 1548: 1530: 1526: 1509: 1507: 1483: 1468: 1461: 1450: 1440: 1436: 1412: 1397: 1390: 1379: 1369: 1365: 1341: 1326: 1319: 1308: 1298: 1294: 1274: 1273: 1244: 1243: 1213: 1208: 1196: 1178: 1166: 1161: 1141: 1136: 1124: 1109: 1097: 1092: 1085: 1084: 1060: 1054: 1053: 1031: 1025: 1024: 993: 992: 978: 977: 959: 948: 933: 926: 914: 899: 897: 885: 864: 853: 838: 831: 819: 804: 802: 790: 777: 754: 751: 750: 732: 721: 706: 699: 687: 672: 670: 658: 637: 626: 611: 604: 592: 577: 575: 563: 550: 527: 524: 523: 505: 494: 479: 472: 460: 445: 443: 431: 410: 399: 384: 377: 365: 350: 348: 336: 323: 300: 291: 290: 267: 262: 261: 237: 224: 211: 194: 188: 187: 184: 153: 97:, specifically 71: 65: 62: 46: 35: 28: 23: 22: 15: 12: 11: 5: 6553: 6551: 6543: 6542: 6537: 6532: 6527: 6522: 6517: 6507: 6506: 6500: 6499: 6497: 6496: 6491: 6489:Predictability 6486: 6481: 6476: 6470: 6468: 6462: 6461: 6459: 6458: 6456:Lai-Sang Young 6453: 6451:James A. Yorke 6448: 6446:Amie Wilkinson 6443: 6438: 6433: 6428: 6423: 6418: 6413: 6408: 6403: 6398: 6393: 6388: 6383: 6381:Henri Poincaré 6378: 6373: 6368: 6363: 6358: 6353: 6348: 6343: 6338: 6333: 6328: 6323: 6318: 6313: 6308: 6303: 6298: 6293: 6287: 6285: 6279: 6278: 6276: 6275: 6270: 6265: 6260: 6255: 6250: 6248:Kicked rotator 6245: 6240: 6235: 6230: 6225: 6220: 6218:Chua's circuit 6214: 6212: 6206: 6205: 6202: 6201: 6199: 6198: 6193: 6188: 6183: 6178: 6173: 6168: 6163: 6158: 6152: 6150: 6147: 6144: 6143: 6141: 6140: 6138:Zaslavskii map 6135: 6133:Tinkerbell map 6130: 6125: 6120: 6115: 6110: 6105: 6100: 6095: 6090: 6085: 6080: 6075: 6070: 6065: 6064: 6063: 6053: 6048: 6043: 6038: 6033: 6028: 6022: 6020: 6017: 6011: 6001: 6000: 5998: 5997: 5992: 5987: 5982: 5980:Ergodic theory 5977: 5972: 5967: 5961: 5959: 5953: 5952: 5937: 5935: 5932: 5931: 5929: 5928: 5923: 5918: 5913: 5908: 5903: 5898: 5892: 5890: 5887: 5884: 5883: 5881: 5880: 5875: 5870: 5865: 5860: 5855: 5850: 5845: 5840: 5835: 5830: 5825: 5820: 5815: 5810: 5805: 5800: 5795: 5790: 5784: 5781: 5780: 5778: 5777: 5772: 5770:Periodic point 5767: 5762: 5757: 5752: 5747: 5742: 5736: 5734: 5731: 5725: 5721: 5720: 5715: 5713: 5712: 5705: 5698: 5690: 5684: 5683: 5670: 5660: 5621: 5620:External links 5618: 5616: 5615: 5571:(10): 114301. 5558: 5552: 5535: 5477: 5428: 5389: 5369:(3): 409–447. 5352: 5347:10.1086/191003 5315: 5302:10.1.1.64.7797 5287:(2): 304–327. 5274: 5241: 5235: 5221: 5218: 5217: 5179: 5177: 5170: 5164: 5161: 5159: 5158: 5148:-body Problem" 5132: 5063: 5016: 4989:(4): 297–300. 4973: 4946:(3): 393–400. 4926: 4891: 4872:(5): 438–442. 4853: 4838: 4820: 4775: 4742: 4691: 4634: 4607:(8): 563–564. 4591: 4584: 4553: 4537: 4492:(1): 225–259. 4474: 4462: 4460: 4459: 4455: 4436: 4426:. 23 June 2021 4411: 4380: 4365: 4342: 4276: 4246: 4217: 4172: 4119: 4076: 4031:(12): 129511. 4015: 3986: 3955:(4): L61–L63. 3935: 3909: 3864:(3): 881–902. 3848: 3784: 3759: 3754: 3747: 3740: 3733: 3726: 3719: 3704: 3698:10.1086/110355 3663: 3644:(6): 113–118. 3624: 3597: 3584: 3556: 3519: 3491: 3475: 3458: 3451: 3420: 3399: 3378: 3376: 3373: 3371: 3370: 3366:3 Body Problem 3362: 3354: 3349: 3344: 3339: 3334: 3329: 3321: 3316: 3311: 3309:Lagrange point 3306: 3304:Gravity assist 3301: 3296: 3290: 3289: 3288: 3285:Physics portal 3274: 3258: 3255: 3147: 3141: 3083:inverse-square 3063: 3060: 3053:neural network 2909:published his 2902: 2899: 2883:n-body problem 2866: 2863: 2809:In the 1970s, 2773:Henri Poincaré 2765: 2761: 2757: 2753: 2749: 2745: 2741: 2715:Leonhard Euler 2710: 2707: 2699: 2698: 2687: 2681: 2678: 2671: 2668: 2663: 2660: 2654: 2648: 2645: 2638: 2635: 2630: 2627: 2621: 2614: 2611: 2576: 2545: 2522: 2519:regularization 2477:Puiseux series 2434: 2431: 2429: 2426: 2413: 2410: 2407: 2402: 2398: 2394: 2391: 2388: 2385: 2380: 2376: 2351: 2347: 2341: 2337: 2333: 2330: 2327: 2324: 2319: 2315: 2309: 2305: 2301: 2298: 2295: 2290: 2285: 2281: 2254: 2247: 2242: 2238: 2231: 2227: 2223: 2220: 2212: 2208: 2204: 2197: 2192: 2188: 2181: 2177: 2173: 2170: 2162: 2158: 2154: 2151: 2143: 2139: 2135: 2130: 2125: 2121: 2114: 2113: 2110: 2103: 2098: 2094: 2087: 2083: 2079: 2076: 2068: 2064: 2060: 2053: 2048: 2044: 2037: 2033: 2029: 2026: 2018: 2014: 2010: 2007: 1999: 1995: 1991: 1986: 1981: 1977: 1970: 1969: 1947: 1927: 1924: 1921: 1918: 1915: 1895: 1890: 1886: 1882: 1877: 1873: 1869: 1849: 1844: 1840: 1836: 1831: 1827: 1823: 1801: 1798: 1795: 1791: 1772:center of mass 1726: 1725: 1684: 1682: 1675: 1669: 1666: 1651: 1640:In this case, 1627: 1619: 1615: 1611: 1605: 1598: 1594: 1586: 1578: 1574: 1570: 1564: 1557: 1553: 1545: 1537: 1533: 1529: 1523: 1516: 1512: 1504: 1497: 1490: 1486: 1481: 1475: 1471: 1465: 1457: 1453: 1447: 1443: 1439: 1433: 1426: 1419: 1415: 1410: 1404: 1400: 1394: 1386: 1382: 1376: 1372: 1368: 1362: 1355: 1348: 1344: 1339: 1333: 1329: 1323: 1315: 1311: 1305: 1301: 1297: 1291: 1288: 1283: 1253: 1229: 1220: 1216: 1211: 1204: 1199: 1193: 1190: 1184: 1181: 1173: 1169: 1164: 1157: 1148: 1144: 1139: 1132: 1127: 1121: 1115: 1112: 1104: 1100: 1095: 1067: 1063: 1038: 1034: 1000: 976: 968: 963: 955: 951: 946: 940: 936: 930: 921: 917: 912: 906: 902: 892: 888: 884: 881: 873: 868: 860: 856: 851: 845: 841: 835: 826: 822: 817: 811: 807: 797: 793: 789: 786: 783: 780: 778: 773: 765: 761: 753: 752: 749: 741: 736: 728: 724: 719: 713: 709: 703: 694: 690: 685: 679: 675: 665: 661: 657: 654: 646: 641: 633: 629: 624: 618: 614: 608: 599: 595: 590: 584: 580: 570: 566: 562: 559: 556: 553: 551: 546: 538: 534: 526: 525: 522: 514: 509: 501: 497: 492: 486: 482: 476: 467: 463: 458: 452: 448: 438: 434: 430: 427: 419: 414: 406: 402: 397: 391: 387: 381: 372: 368: 363: 357: 353: 343: 339: 335: 332: 329: 326: 324: 319: 311: 307: 299: 298: 274: 270: 249: 244: 240: 236: 231: 227: 223: 218: 214: 210: 207: 201: 197: 183: 180: 83:center of mass 73: 72: 49: 47: 40: 26: 24: 18:3-body problem 14: 13: 10: 9: 6: 4: 3: 2: 6552: 6541: 6538: 6536: 6533: 6531: 6528: 6526: 6523: 6521: 6518: 6516: 6513: 6512: 6510: 6495: 6492: 6490: 6487: 6485: 6484:Edge of chaos 6482: 6480: 6477: 6475: 6472: 6471: 6469: 6463: 6457: 6454: 6452: 6449: 6447: 6444: 6442: 6441:Marcelo Viana 6439: 6437: 6434: 6432: 6431:Audrey Terras 6429: 6427: 6426:Floris Takens 6424: 6422: 6419: 6417: 6414: 6412: 6409: 6407: 6404: 6402: 6399: 6397: 6394: 6392: 6389: 6387: 6384: 6382: 6379: 6377: 6374: 6372: 6369: 6367: 6364: 6362: 6359: 6357: 6354: 6352: 6349: 6347: 6344: 6342: 6339: 6337: 6334: 6332: 6329: 6327: 6326:Celso Grebogi 6324: 6322: 6319: 6317: 6314: 6312: 6309: 6307: 6306:Chen Guanrong 6304: 6302: 6299: 6297: 6294: 6292: 6291:Michael Berry 6289: 6288: 6286: 6280: 6274: 6271: 6269: 6266: 6264: 6261: 6259: 6256: 6254: 6251: 6249: 6246: 6244: 6241: 6239: 6236: 6234: 6231: 6229: 6226: 6224: 6221: 6219: 6216: 6215: 6213: 6207: 6197: 6194: 6192: 6189: 6187: 6184: 6182: 6179: 6177: 6174: 6172: 6169: 6167: 6166:Lorenz system 6164: 6162: 6159: 6157: 6154: 6153: 6151: 6145: 6139: 6136: 6134: 6131: 6129: 6126: 6124: 6121: 6119: 6116: 6114: 6113:Langton's ant 6111: 6109: 6106: 6104: 6101: 6099: 6096: 6094: 6091: 6089: 6088:Horseshoe map 6086: 6084: 6081: 6079: 6076: 6074: 6071: 6069: 6066: 6062: 6059: 6058: 6057: 6054: 6052: 6049: 6047: 6044: 6042: 6039: 6037: 6034: 6032: 6029: 6027: 6024: 6023: 6021: 6015: 6012: 6009: 6002: 5996: 5993: 5991: 5988: 5986: 5985:Quantum chaos 5983: 5981: 5978: 5976: 5973: 5971: 5968: 5966: 5963: 5962: 5960: 5954: 5949: 5945: 5941: 5927: 5924: 5922: 5919: 5917: 5914: 5912: 5909: 5907: 5904: 5902: 5899: 5897: 5894: 5893: 5891: 5885: 5879: 5876: 5874: 5871: 5869: 5866: 5864: 5861: 5859: 5856: 5854: 5851: 5849: 5846: 5844: 5841: 5839: 5836: 5834: 5831: 5829: 5826: 5824: 5821: 5819: 5816: 5814: 5811: 5809: 5806: 5804: 5801: 5799: 5796: 5794: 5793:Arnold tongue 5791: 5789: 5786: 5785: 5782: 5776: 5773: 5771: 5768: 5766: 5763: 5761: 5758: 5756: 5753: 5751: 5748: 5746: 5743: 5741: 5738: 5737: 5735: 5729: 5726: 5722: 5718: 5711: 5706: 5704: 5699: 5697: 5692: 5691: 5688: 5681: 5677: 5674: 5671: 5668: 5664: 5661: 5656: 5651: 5647: 5643: 5639: 5635: 5634: 5629: 5624: 5623: 5619: 5612: 5608: 5604: 5600: 5596: 5592: 5588: 5584: 5579: 5574: 5570: 5566: 5565: 5559: 5555: 5549: 5544: 5543: 5536: 5526:on 2018-10-08 5522: 5518: 5514: 5510: 5506: 5502: 5498: 5494: 5490: 5483: 5478: 5474: 5470: 5466: 5462: 5458: 5454: 5449: 5444: 5440: 5436: 5435: 5429: 5425: 5421: 5417: 5413: 5409: 5405: 5401: 5397: 5396: 5390: 5385: 5380: 5376: 5372: 5368: 5364: 5363: 5358: 5353: 5348: 5343: 5339: 5335: 5331: 5327: 5326: 5321: 5316: 5312: 5308: 5303: 5298: 5294: 5290: 5286: 5282: 5281: 5275: 5271: 5267: 5262: 5257: 5254:: 1088–1100. 5253: 5249: 5248: 5242: 5238: 5232: 5228: 5223: 5222: 5214: 5211: 5203: 5193: 5192:editing guide 5187: 5183: 5178: 5169: 5168: 5162: 5155: 5154: 5149: 5147: 5141: 5136: 5133: 5128: 5124: 5120: 5116: 5112: 5108: 5104: 5100: 5096: 5092: 5087: 5082: 5079:(6): 065901. 5078: 5074: 5067: 5064: 5059: 5055: 5051: 5047: 5043: 5039: 5035: 5031: 5027: 5020: 5017: 5012: 5008: 5004: 5000: 4996: 4992: 4988: 4984: 4977: 4974: 4969: 4965: 4961: 4957: 4953: 4949: 4945: 4941: 4937: 4930: 4927: 4922: 4918: 4914: 4910: 4906: 4902: 4895: 4892: 4887: 4883: 4879: 4875: 4871: 4867: 4860: 4858: 4854: 4849: 4845: 4841: 4835: 4831: 4824: 4821: 4816: 4812: 4808: 4804: 4799: 4794: 4790: 4786: 4779: 4776: 4763: 4759: 4758: 4753: 4746: 4743: 4738: 4734: 4729: 4724: 4719: 4714: 4710: 4706: 4702: 4695: 4692: 4687: 4683: 4678: 4673: 4669: 4665: 4661: 4657: 4653: 4649: 4645: 4638: 4635: 4630: 4626: 4622: 4618: 4614: 4610: 4606: 4602: 4595: 4592: 4587: 4581: 4577: 4573: 4566: 4565: 4557: 4554: 4550: 4546: 4541: 4538: 4533: 4529: 4525: 4521: 4517: 4513: 4509: 4505: 4500: 4495: 4491: 4487: 4486: 4478: 4475: 4471: 4466: 4463: 4456: 4453: 4452: 4450: 4446: 4440: 4437: 4425: 4421: 4415: 4412: 4401: 4397: 4393: 4392: 4384: 4381: 4376: 4372: 4368: 4362: 4358: 4351: 4349: 4347: 4343: 4331: 4327: 4323: 4319: 4315: 4311: 4306: 4301: 4298:(3): 031020. 4297: 4293: 4292: 4287: 4280: 4277: 4265: 4261: 4257: 4250: 4247: 4236:on 2022-11-17 4235: 4231: 4227: 4221: 4218: 4213: 4209: 4205: 4201: 4196: 4191: 4187: 4183: 4176: 4173: 4168: 4164: 4160: 4156: 4152: 4148: 4143: 4138: 4134: 4130: 4129:New Astronomy 4123: 4120: 4114: 4109: 4104: 4099: 4095: 4091: 4087: 4080: 4077: 4072: 4068: 4064: 4060: 4056: 4052: 4048: 4044: 4039: 4034: 4030: 4026: 4019: 4016: 4004: 3997: 3990: 3987: 3981: 3976: 3972: 3968: 3963: 3958: 3954: 3950: 3946: 3939: 3936: 3931: 3927: 3920: 3913: 3910: 3905: 3901: 3897: 3893: 3889: 3885: 3881: 3877: 3872: 3867: 3863: 3859: 3852: 3849: 3838:on 2018-10-08 3834: 3830: 3826: 3822: 3818: 3814: 3810: 3806: 3802: 3795: 3788: 3785: 3773: 3766: 3764: 3760: 3753: 3746: 3739: 3736:(0) = (0,0); 3732: 3725: 3718: 3714: 3708: 3705: 3699: 3694: 3690: 3686: 3682: 3678: 3674: 3667: 3664: 3659: 3655: 3651: 3647: 3643: 3639: 3635: 3628: 3625: 3620: 3616: 3612: 3608: 3601: 3598: 3594: 3588: 3585: 3574: 3570: 3563: 3561: 3557: 3552: 3548: 3543: 3538: 3534: 3530: 3523: 3520: 3508:. 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