digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Three-body problem has two distinguishable meanings in physics and classical mechanics:

1. In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton's laws of motion and of universal gravitation).
2. In an extended modern sense a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. Typically, all three particles are considered as point masses, neglecting their shape and internal structure, and the interaction among them is a scalar potential such as gravity or electromagnetism.[citation needed]

Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun.[1]

## History

The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his 'Principia' (Philosophiæ Naturalis Principia Mathematica). 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 the Earth and the Sun.

During the second quarter of the eighteenth century, the problem of improving the accuracy of the lunar theory came to be of topical interest. The topicality arose mainly because it was perceived that the results should be applicable to navigation, that is, to the development of a method for determining geographical longitude at sea. Following Newton's work, it was appreciated that at least a major part of the problem in lunar theory consisted in evaluating the perturbing effect of the Sun on the motion of the Moon around the Earth.

Jean d'Alembert and Alexis Clairaut, who developed a longstanding rivalry, both attempted to analyze the problem in some degree of generality, and by the use of differential equations to be solved by successive approximations. They submitted their competing first analyses to the Académie Royale des Sciences in 1747.[2]

It was in connection with these researches, in Paris, in the 1740s, that the name "three-body problem" (Problème des Trois Corps) began to be commonly used. An account published in 1761 by Jean d'Alembert indicates that the name was first used in 1747.[3]

In 1887, mathematicians Ernst Bruns [4] and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases.[5]

## Examples

The circular restricted three-body problem is a valid approximation of elliptical orbits found in the Solar System, 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 (shown as blue lines) indicating that the forces are in balance there.

### Gravitational systems

A prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In most cases such a system can be factorized, considering the movement of the complex system (planet and satellite) around a star as a single particle; then, considering the movement of the satellite around the planet, neglecting the movement around the star. In this case, the problem is simplified to the two-body problem. However, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation.

A three-body problem also arises from the situation of a spacecraft and two relevant celestial bodies, e.g. the Earth and the Moon, such as when considering a free return trajectory around the Moon, or other trans-lunar injection. While a spaceflight involving a gravity assist tends to be at least a four-body problem (spacecraft, Earth, Sun, Moon), once far away from the Earth when Earth's gravity becomes negligible, it is approximately a three-body problem.

#### Circular restricted three-body problem

In the circular restricted three-body problem two massive bodies move in circular orbits around their common center of mass, and the third mass is small and moves in the same plane.[6] 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 orbit around them, for instance on a horseshoe orbit. It can be useful to consider the effective potential.

#### Constant-pattern solutions

Lagrange, tackling the general three-body problem, considered the behaviour of the distances between the bodies, without finding a general solution. But from his numerous equations he discovered two classes of constant-pattern solutions : collinear, in which one of the distances is the sum of the other two, and equiangular, in which the three distances are equal. Those classes yield what are now called L1, L2, L3 and L4, L5.

In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions, bringing the total number of families of repetitive motion to 16. One of the 16 families is a figure-eight pattern discovered in 1993 by physicist Cris Moore at the Santa Fe Institute.[5]

### Atomic systems

Due to the small value of the fine-structure constant, various atomic systems, such as atoms of helium or the helium-like ions can be described as three-body systems;[citation needed] however, at high atomic numbers, the velocities become relativistic and consequently, the approximation becomes inaccurate. In the case of the helium atom or helium-like ions, the system is determined by the mass of the nucleus, mass of the electron, and the Coulomb interaction between them.[citation needed]

In addition, some properties of simple molecules can be described, assuming the fast movement of electrons (which are many orders of magnitude lighter than nuclei); then, the electrons determine some effective potential, and the movement of atoms can be described with this potential. In this sense, the triatomic molecule (for example, water, or the carbon dioxide) can be treated as a three-body problem.[citation needed] This description is valid at weak excitations (for example, at room temperature), and can be used for estimating the thermal capacities of gases; however, it is important to determine the number of vibrational degrees of freedom at a given temperature.[citation needed]

In the 21st century, experiments with atomic traps and molecular traps enhance the possibilities to deal with three-body systems.[citation needed]

Upon excitation with short pulses, during the short time after the excitation, such systems may show trajectories and other attributes typical of classical mechanics.[citation needed]

## Classical versus quantum mechanics

Physicist Vladimir Krivchenkov used the three-body problem as an example, showing the simplicity of quantum mechanics in comparison to classical mechanics. The quantum three-body problem is studied in university courses of quantum mechanics;[7] in particular, the energy of the ground state and the first excited states can be estimated by hand, even without the use of computers, using perturbation theory.[citation needed] As for classical mechanics, the variety of divergent trajectories with various Lyapunov exponents[citation needed] makes the problem too difficult for undergraduate courses.

The three-body system is one of the simplest classical mechanical systems that allows for unstable trajectories. In the case of gravitating masses, one of the questions of the three-body problem is: For some given probability distribution over initial conditions, what is the probability that during some time t, two particles get close enough, providing the energy that would allow the third particle to leave the system?[citation needed]

In the case of quantum mechanics, the main part of the three-body problem refers to the finding the eigenstates and their energies.[citation needed]. For a special case of the quantum three-body problem known as the hydrogen molecular ion, the eigenenergies are solvable analytically (see discussion in quantum mechanical version of Euler's three-body problem) in terms a generalization of the Lambert W function.

## n-body problem

The three-body problem is a special case of the n-body problem, which describes how n objects will move under one of the physical forces, such as gravity. These problems have a global analytical solution in form of a convergent power series, as it was proven by Sundman for n = 3 and by 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;[8] 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 star(s), 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.

## Notes

1. ^ "Historical Notes: Three-Body Problem". Retrieved December 2010.
2. ^ 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).
3. ^ Jean 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.
4. ^ J J O'Connor and E F Robertson (August 2006). "Bruns biography". Retrieved 04.04.2013. Text " University of St. Andrews, Scotland " ignored (help)
5. ^ a b Jon Cartwright (8 March 2013). "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem". Science. Retrieved 04.04.2013.
6. ^ Restricted Three-Body Problem, Science World.
7. ^ I. I. Gol’dman and V. D. Krivchenkov. Problems in Quantum Mechanics.. 3rd ed. Mineola, N.Y.: Dover Publications, 2006. 288 pp.
8. ^ Diacu, Florin. "The Solution of the n-body Problem*", The Mathematical Intelligencer, 1996.

## References

• Aarseth S. J., Gravitational n-Body Simulations, 2003, Cambridge University Press.
• Bagla J. S., "Cosmological N-body simulation: Techniques, scope and status", 2005, Current Science.
• Chambers J. E., Wetherill G. W., Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions, 1998, Academic Press.
• Efstathiou G., Davis M., White S. D. M., Frenk C. S., "Numerical techniques for large cosmological N-body simulations", 1985, ApJ.
• Hulkower Neal D., "The Zero Energy Three Body Problem", Indiana University Mathematics Journal 27 (1978) pp. 409–447.
• Hulkower Neal D., "Central Configurations and Hyperbolic-Elliptic Motion in the Three-Body Problem", Celestial Mechanics 21 (1980) pp. 37–41.
• Šuvakov, Milovan; Dmitrašinović, V. (2013). "Three Classes of Newtonian Three-Body Planar Periodic Orbits". arXiv:1303.0181.
 1000000 videos foundNext >
 Voyager and the Three Body ProblemFrom the BBC Horizon documentary Voyager: To the Final Frontier, Dallas Campbell tells the story of how mathematician Michael Minovitch solved the Three Body... Restricted 3-Body Problem With Osculating OrbitsThe osculating orbit of an object in space is the gravitational Kepler orbit that it would have if perturbations were not present. For each body exactly one ... Three-body problem sonifiedSound is generated when distance between two 'planets' is minimal, one note of C major triad is assigned to each pair. You can support me here: http://www.pa... 3 Body problem: A stable binary systemA restricted 3 body problem. Solution obtained numericaly with the RK4 method. A restricted three body problem solutionA binary system with a planet in the center of mass. Is is verys stable until a very small numerical error make the planet get a chaotic orbit. 47 points in Three Body ProblemFinally beat the developer! (by one point) http://www.kongregate.com/games/roBurky/three-body-problem. Pythagorean 3-Body Problem With Osculating OrbitsThe osculating orbit of an object in space is the gravitational Kepler orbit that it would have if perturbations were not present. For each body exactly one ... The Klein Four Group -- Three Body ProblemThree Body Problem from the Klein Four Group. Their website: http://www.kleinfour.com/ Mod-01 Lec-19 Lecture-19-Three Body Problem (Contd...7)Space Flight Mechanics by Dr.Manoranjan Sinha,Department of Aerospace Engineering ,IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in. Three Body Problem Gameplay Video - TVGTThree Body Problem Gameplay Video, played by The Videogame Tryer. Free download: http://www.roburky.co.uk/games/three-body-problem/
 1000000 videos foundNext >
 2 news items
 Osservatorio Balcani e Caucaso Serbia: solving the three-body problem Osservatorio Balcani e Caucaso Thu, 02 May 2013 00:58:35 -0700 Known as “the three-body problem”, the enigma is one of the oldest in the history of physics and is still of vital importance today. Indeed, a better understanding of the effects of the gravitational attraction amongst three or more celestial bodies ... Chaos at fifty PhysicsToday.org Tue, 30 Apr 2013 13:10:43 -0700 But researchers had experienced close encounters with the phenomenon as early as the late 1880s, beginning with Henri Poincaré's studies of the three-body problem in celestial mechanics. Poincaré observed that in such systems “it may happen that small ...
 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter