Three-body problem

File:Three-body Problem Animation with COM.gif

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 orbiting each other in space and then calculate their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.<ref name="PrincetonCompanion">Template:Cite encyclopedia</ref>

Unlike the two-body problem, the three-body problem has no general closed-form solution, meaning there is no equation that always solves it.<ref name="PrincetonCompanion"/> When three bodies orbit each other, the resulting dynamical system is 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|Template:Mvar-body problem]]. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun.<ref name="first">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three particles.

Mathematical description

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

<math display="block">\begin{align}

\ddot\mathbf{r}_1 &= -G m_2 \frac{ \left( \mathbf{r}_1 - \mathbf{r}_2 \right) }{\ \left|\mathbf{r}_1 - \mathbf{r}_2 \right|^3 } - G m_3 \frac{ \left( \mathbf{r}_1 - \mathbf{r}_3 \right)}{\ \left|\mathbf{r}_1 - \mathbf{r}_3\right|^3}\ , \\
\ddot\mathbf{r}_2 &= -G m_3 \frac{ \left( \mathbf{r}_2 - \mathbf{r}_3 \right) }{\ \left|\mathbf{r}_2 - \mathbf{r}_3 \right|^3 } - G m_1 \frac{ \left( \mathbf{r}_2 - \mathbf{r}_1 \right) }{\ \left|\mathbf{r}_2 - \mathbf{r}_1 \right|^3 }\ , \\
\ddot\mathbf{r}_3 &= -G m_1 \frac{ \left( \mathbf{r}_3 - \mathbf{r}_1 \right) }{\ \left|\mathbf{r}_3 - \mathbf{r}_1 \right|^3 } - G m_2 \frac{ \left( \mathbf{r}_3 - \mathbf{r}_2 \right) }{\ \left|\mathbf{r}_3 - \mathbf{r}_2 \right|^3 } ~.

\end{align}</math> where <math>\ G\ </math> is the gravitational constant. As astronomer Juhan Frank describes, "These three second-order vector differential equations are equivalent to 18 first order scalar differential equations."<ref>This explicit report on the vector presentation appears drawn, through generalisation, from the <math>\ \ddot\mathbf{r}_i\ </math> expression presented by Cambridge-educated astronomer Juhan Frank of LSU, as presented in his class notes for Physics 7221 in 2006, see {{#invoke:citation/CS1|citation |CitationClass=web

}}.Template:Better source</ref>Template:Better source As June Barrow-Green notes with regard to an alternative presentation, if

<math>P_i</math> represent three particles with masses <math>m_i</math>, distances <math>\ P_i P_j = r_{ij}\ ,</math> and coordinates <math>\ q_{ij}\ </math> <math>\ (i,j = 1,2,3)\ </math> in an inertial coordinate system ... the problem is described by nine second-order differential equations.<ref name="Barrow-Green1997">For a more general discussion of the presentation of these equations in non-vector formats not explicitly related to the presentation in text, see the authoritative Template:Cite book</ref>Template:Rp

The problem can also be stated equivalently in the Hamiltonian formalism, in which case it is described by a set of 18 first-order differential equations, one for each component of the positions <math>\ \mathbf{r}_i\ </math> and momenta <math>\ \mathbf{p}_i\ </math>:Template:Cn<ref>For a related presentation of the Hamiltonian, which chooses units and presentation to simplify the maths, see Barrow-Green, p. 8, op. cit.</ref>

<math display="block"> \frac{\mathrm d\ \mathbf{r}_i}{\mathrm d\ t} = \frac{\partial\ \mathcal{H} }{ \partial\ \mathbf{p}_i}\ , \qquad \frac{ \mathrm d\ \mathbf{p}_i }{ \mathrm d\ t } = -\frac{\partial\ \mathcal{H}}{\partial\ \mathbf{r}_i }\ , </math>

where <math>\mathcal{H}</math> is the Hamiltonian:Template:Cn

<math display="block"> \mathcal{H}\ =\ -\frac{G m_1 m_2}{\left|\mathbf{r}_1 - \mathbf{r}_2\right|}\ -\ \frac{G m_2 m_3}{\left|\mathbf{r}_3 - \mathbf{r}_2\right|}\ -\ \frac{G m_3 m_1}{\left|\mathbf{r}_3 - \mathbf{r}_1\right|}\ +\ \frac{\left|\mathbf{p}_1\right|^2}{2m_1}\ +\ \frac{\left|\mathbf{p}_2\right|^2}{2m_2}\ +\ \frac{\left|\mathbf{p}_3\right|^2}{2m_3} ~. </math>

In this case, <math>\mathcal{H}</math> is simply the total energy of the system, gravitational plus kinetic.Template:Cn

Restricted three-body problemTemplate:AnchorTemplate:Anchor

File:Restricted Three-Body Problem - Energy Potential Analysis.png

The circular restricted three-body problemTemplate:Clarify is a valid approximation of elliptical orbits found in the Solar System,Template: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.Template:Citation needed

In the restricted three-body problem formulation, in the description of Barrow-Green,<ref name="Barrow-Green1997"/>Template:Rp

two... bodies revolve around their centre of mass in circular orbits under the influence of their mutual gravitational attraction, and... form a two body system... [whose] motion is known. A third body (generally known as a planetoid), assumed massless with respect to the other two, moves in the plane defined by the two revolving bodies and, while being gravitationally influenced by them, exerts no influence of its own.<ref name="Barrow-Green1997"/>Template:Rp

Per Barrow-Green, "[t]he problem is then to ascertain the motion of the third body."<ref name="Barrow-Green1997"/>Template:Rp

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.Template:What (That is, it is useful to consider the effective potential.Template:WhatTemplate:Says who) 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.Template:Cn

The restricted three-body problem is easier to analyze theoretically than the full problem. It is of practical interest as well since it accurately describes many real-world problems, the most important example being the Earth–Moon–Sun system. For these reasons, it has occupied an important role in the historical development of the three-body problem.<ref name="first2024">Template:Cite journal</ref>

Mathematically, the problem is stated as follows.Template:Cn Let <math>\ m_1, m_2\ </math> be the masses of the two massive bodies, with (planar) coordinates <math>\ (x_1, y_1)\ </math> and <math>\ (x_2, y_2)\ ,</math> and let <math>\ (x, y)\ </math> be the coordinates of the planetoid. For simplicity, choose units such that the distance between the two massive bodies, as well as the gravitational constant, are both equal to <math>\ 1 ~.</math> Then, the motion of the planetoid is given by:Template:Cn

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

where <math>\ r_i \equiv \sqrt{(x - x_i)^2 + (y - y_i)^2 \;} ~.</math>Template:Cn In this form the equations of motion carry an explicit time dependence through the coordinates <math>\ x_i(t), y_i(t)\ ;</math>Template:Cn however, if the two bodies are uniformly rotating, this time dependence can be removed through a transformation to their rotating reference frame, which simplifies any subsequent analysis.Template:Or<ref>Note, the following source does not state that the "time dependence can be removed through a transformation to a rotating reference frame." For a related but distinct presentation of the restricted three-body problem—featuring the Jacobi integral for the "energy of <math>\ m_3\ </math> in the co-rotating (non-inertial) frame of the primaries"—see Template:Cite journal</ref>

Solutions

General solution

File:3bodyproblem.gif

While a system of 3 bodies interacting gravitationally is chaotic, a system of 3 bodies interacting elastically is not.Template:Clarify

There is no general closed-form solution to the three-body problem.<ref name="PrincetonCompanion"/> In other words, it does not have a general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.<ref name=13solutions>Template:Cite journal</ref>

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 Template:Math.<ref>Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.</ref> This series converges for all real Template:Mvar, 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 Template:Math vanishes. By restricting the initial data to Template:Math, he removed all real singularities from the transformed equations for the three-body problem.
3. Showing that if Template:Math, 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 Template:Math) 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 Template:Math (the new variable after the regularization) and if Template:Math,Template:Clarify then this map is given by <math display="block">\sigma = \frac{e^\frac{\pi s}{2\beta} - 1}{e^\frac{\pi s}{2\beta} + 1}.</math>

This finishes the proof of Sundman's theorem.

The corresponding series converges extremely slowly. That is, obtaining a value of meaningful precision requires so many terms that this solution is of little practical use. Indeed, in 1930, David Beloriszky calculated that if Sundman's series were to be used for astronomical observations, then the computations would involve at least 10^Template:Val terms.<ref>Template:Cite journal</ref>

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<ref name="Burrau">Template:Cite journal</ref> further investigated this problem in 1913. In 1967 Victor Szebehely and C. Frederick Peters established eventual escape of the lightest body for this problem using numerical integration, while at the same time finding a nearby periodic solution.<ref>Template:Cite journal</ref>

File:Three body problem figure-8 orbit animation.gif

An animation of the figure-8 solution to the three-body problem over a single period T ≃ 6.3259<ref>Here the gravitational constant G has been set to 1, and the initial conditions are 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).</ref>

File:5 4 800 36 downscaled.gif

20 examples of periodic solutions to the three-body problem

In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Hénon–Hadjidemetriou family. In this family, the three objects all have the same mass and can exhibit both retrograde and direct forms. In some of Broucke's solutions, two of the bodies follow the same path.<ref name="TBG">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In 1993, physicist Cris Moore at the Santa Fe Institute found a zero angular momentum solution with three equal masses moving around a figure-eight shape.<ref>Template:Cite journal</ref> In 2000, mathematicians Alain Chenciner and Richard Montgomery proved its formal existence.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> The solution has been shown numerically to be stable for small perturbations of the mass and orbital parameters, which makes it possible for such orbits to be observed in the physical universe. But it has been argued that this is unlikely since the domain of stability is small. For instance, the probability of a binary–binary scattering eventTemplate:Clarify resulting in a figure-8 orbit has been estimated to be a small fraction of a percent.<ref>Template:Cite journal</ref>

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<ref name=13solutions/><ref name="TBG"/>

In 2015, physicist Ana Hudomal discovered 14 new families of solutions for the equal-mass zero-angular-momentum three-body problem.<ref>Template:Cite journal</ref>

In 2017, researchers Xiaoming Li and Shijun Liao found 669 new periodic orbits of the equal-mass zero-angular-momentum three-body problem.<ref>Template:Cite journal</ref> This was followed in 2018 by an additional 1,223 new solutions for a zero-angular-momentum system of unequal masses.<ref>Template:Cite journal</ref>

In 2018, Li and Liao reported 234 solutions to the unequal-mass "free-fall" three-body problem.<ref>Template:Cite journal</ref> The free-fall formulation starts with all three bodies at rest. Because of this, the masses in a free-fall configuration do not orbit in a closed "loop", but travel forward and backward along an open "track".

In 2023, Ivan Hristov, Radoslava Hristova, Dmitrašinović and Kiyotaka Tanikawa published a search for "periodic free-fall orbits" three-body problem, limited to the equal-mass case, and found 12,409 distinct solutions.<ref>Template:Cite journal</ref>

Numerical approaches

Using a computer, the problem may be solved to arbitrarily high precision using numerical integration. There have been attempts of creating computer programs that numerically solve the three-body problem (and by extension, the n-body problem) involving both electromagnetic and gravitational interactions, and incorporating modern theories of physics such as special relativity.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In addition, using the theory of random walks, an approximate probability of different outcomes may be computed.<ref>Template:Cite news</ref><ref>Template:Cite journal</ref>

History

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Philosophiæ Naturalis Principia Mathematica, in which Newton attempted to figure out if any long term stability is possible especially for such a system like that of the Earth, the Moon, and the Sun, after having solved the two-body problem.<ref>Template:Cite book</ref> Guided by major Renaissance astronomers Nicolaus Copernicus, Tycho Brahe and Johannes Kepler, Newton introduced later generations to the beginning of the gravitational three-body problem.<ref name=":1">Template:Cite book</ref> In Proposition 66 of Book 1 of the Principia, and its 22 Corollaries, Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions. In Propositions 25 to 35 of Book 3, Newton also took the first steps in applying his results of Proposition 66 to the lunar theory, the motion of the Moon under the gravitational influence of Earth and the Sun.<ref>Template:Cite book</ref> Later, this problem was also applied to other planets' interactions with the Earth and the Sun.<ref name=":1" />

The physical problem was first addressed by Amerigo Vespucci and subsequently by Galileo Galilei, as well as Simon Stevin, but they did not realize what they contributed. Though Galileo determined that the speed of fall of all bodies changes uniformly and in the same way, he did not apply it to planetary motions.<ref name=":1" /> Whereas in 1499, Vespucci used knowledge of the position of the Moon to determine his position in Brazil.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>The 1747 memoirs of both parties can be read in the volume of Histoires (including Mémoires) of the Académie Royale des Sciences for 1745 (belatedly published in Paris in 1749) (in French):

Clairaut: "On the System of the World, according to the principles of Universal Gravitation" (at pp. 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).</ref> It was in connection with their research, in Paris during the 1740s, that the name "three-body problem" (Template:Langx) began to be commonly used. An account published in 1761 by Jean le Rond d'Alembert indicates that the name was first used in 1747.<ref>Jean le Rond d'Alembert, in a paper of 1761 reviewing the mathematical history of the problem, mentions that Euler had given a method for integrating a certain differential equation "in 1740 (seven years before there was question of the Problem of Three Bodies)": see d'Alembert, "Opuscules Mathématiques", vol. 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.</ref>

From the end of the 19th century to early 20th century, the approach to solve the three-body problem with the usage of short-range attractive two-body forces was developed by scientists, which offered P. F. Bedaque, H.-W. Hammer and U. van Kolck an idea to renormalize the short-range three-body problem, providing scientists a rare example of a renormalization group limit cycle at the beginning of the 21st century.<ref name=":0">Template:Cite journal</ref> George William Hill worked on the restricted problem in the late 19th century with an application of motion of Venus and Mercury.<ref>"Coplanar Motion of Two Planets, One Having a Zero Mass". Annals of Mathematics, Vol. III, pp. 65–73, 1887.</ref>

At the beginning of the 20th century, Karl Sundman approached the problem mathematically and systematically by providing a functional theoretical proof to the problem valid for all values of time. It was the first time scientists theoretically solved the three-body problem. However, because there was not a qualitative enough solution of this system, and it was too slow for scientists to practically apply it, this solution still left some issues unresolved.<ref>Template:Cite book</ref> In the 1970s, implication to three-body from two-body forces had been discovered by V. Efimov, which was named the Efimov effect.<ref>Template:Cite journal</ref>

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.<ref>Template:Cite journal</ref>

In 2019, Breen et al. announced a fast neural network solver for the three-body problem, trained using a numerical integrator.<ref>Template:Cite journal</ref>

In September 2023, several possible solutions have been found to the problem according to reports.<ref name="SA-20230923">Template:Cite news</ref><ref name="ARX-2023">Template:Cite journal</ref>

Other problems involving three bodies

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

A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.<ref>Template:Cite book</ref>

In both classical and quantum mechanics, however, there exist nontrivial interaction laws besides the inverse-square force that do lead to exact analytic three-body solutions. One such model consists of a combination of harmonic attraction and a repulsive inverse-cube force.<ref name="Crandall1984">Template:Cite journal</ref> This model is considered nontrivial since it is associated with a set of nonlinear differential equations containing singularities (compared with, e.g., harmonic interactions alone, which lead to an easily solved system of linear differential equations). In these two respects it is analogous to (insoluble) models having Coulomb interactions, and as a result has been suggested as a tool for intuitively understanding physical systems like the helium atom.<ref name="Crandall1984"/><ref>Template:Cite journal</ref>

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,<ref>Template:Cite journal</ref> while at least four vortices are required to obtain chaotic behavior.<ref>Template:Cite journal</ref> One can draw parallels between the motion of a passive tracer particle in the velocity field of three vortices and the restricted three-body problem of Newtonian mechanics.<ref>Template:Cite journal</ref>

The gravitational three-body problem has also been studied using general relativity. Physically, a relativistic treatment becomes necessary in systems with very strong gravitational fields, such as near the event horizon of a black hole. However, the relativistic problem is considerably more difficult than in Newtonian mechanics, and 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.<ref>Template:Cite journal</ref>

Template:Mvar-body problem

The three-body problem is a special case of the [[n-body problem|Template:Mvar-body problem]], which describes how Template:Mvar 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 Template:Math and by Qiudong Wang for Template:Math (see [[n-body problem|Template:Mvar-body problem]] for details). However, the Sundman and Wang series converge so slowly that they are useless for practical purposes;<ref>Florin Diacu. "The Solution of the n-body Problem", The Mathematical Intelligencer, 1996.</ref> therefore, it is currently necessary to approximate solutions by numerical analysis in the form of numerical integration or, for some cases, classical trigonometric series approximations (see [[N-body simulation|Template:Mvar-body simulation]]). Atomic systems, e.g. atoms, ions, and molecules, can be treated in terms of the quantum Template:Mvar-body problem. Among classical physical systems, the Template:Mvar-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 Template:Mvar-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|Template:Mvar-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)

Further reading

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External links

• Template:Cite journal
• 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)