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==Examples==
==Examples==
Due to the small value of the [[fine-structure constant]], various atomic systems can be described as 3-body systems:{{Citation needed|date=July 2009}} Atoms of [[helium]] or the [[helium-like ion]]s for example; however, at high [[atomic number]]s, the velocities become [[Theory of relativity|relativistic]] and consequently the approximation becomes inaccurate.
Due to the small value of the [[fine-structure constant]], various atomic systems, such as atoms of [[helium]] or the [[helium-like ion]]sn can be described as 3-body systems;{{Citation needed|date=July 2009}} however, at high [[atomic number]]s, the velocities become [[Theory of relativity|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 [[atomic nucleus|nucleus]], [[electron mass|mass of the electron]], and the [[Coulomb interaction]] between them.{{Citation needed|date=July 2009}}

In the case of the helium atom or helium-like ions, the system is determined by the mass of the [[atomic nucleus|nucleus]], [[electron mass|mass of the electron]], and the [[Coulomb interaction]] between them.{{Citation needed|date=July 2009}}


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 3-atomic molecule (for example, [[water]], or the [[carbon dioxide]]) can be treated as a 3-body problem.{{Citation needed|date=July 2009}}
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 3-atomic molecule (for example, [[water]], or the [[carbon dioxide]]) can be treated as a 3-body problem.{{Citation needed|date=July 2009}}
This description is valid at weak excitations (for example, at room temperature), and can be used for estimating the [[thermal capacities]] of gases; the crucial thing to determine is, how many [[vibrational degrees of freedom]] are excited at a given temperature.{{Citation needed|date=July 2009}}
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|date=July 2009}}


In the 21st century, experiments with [[atomic trap]]s and [[molecular trap]]s enhance the possibilities to deal with 3-body systems.{{Citation needed|date=July 2009}}
In the 21st century, experiments with [[atomic trap]]s and [[molecular trap]]s enhance the possibilities to deal with 3-body systems.{{Citation needed|date=July 2009}}

Revision as of 03:46, 28 January 2012

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' (Philosophiae 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]

Approximations

The energy E of movement is assumed to be small compared to their mass, allowing one to describe the bodies with non-relativistic mechanics. This implies that all the movement refers to velocities small compared to the speed of light c.

In quantum field theory, at high speed, the creation and annihilation of particles becomes possible, so, it is not possible to keep the number of particles constant.

In such a way, the 3-body problem requires a certain class of approximations.

Examples

Due to the small value of the fine-structure constant, various atomic systems, such as atoms of helium or the helium-like ionsn can be described as 3-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 3-atomic molecule (for example, water, or the carbon dioxide) can be treated as a 3-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 3-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]

Another example of a classical 3-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 (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 2-body problem. However, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation.

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.[4] 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.

Classical versus quantum mechanics

Physicist Vladimir Krivchenkov used the 3-body problem as an example, showing the simplicity of quantum mechanics in comparison to classical mechanics. The quantum 3-body problem is studied in university courses of quantum mechanics;[5] 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 3-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 3-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 3-body problem refers to the finding the eigenstates and their energies.[citation needed]. For a special case of the quantum 3-body problem known as the Hydrogen Molecular ion, the eigenenergies are solvable analytically (see discussion in quantum mechanical version of Euler's 3-body problem) in terms a generalization of the Lambert W function.

n-body problem

The 3-body problem is a special case of the n-body problem. n-body problems deal with the question of 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 convergence of the Sundman and Wang series is so slow that they are useless for practical purposes, and that is why, for the time being, approximate solution by numerical analysis in the form of numerical integration (or, for some cases, classical trigonometric series approximations) is unavoidable, see n-body simulation. Atomic systems (atoms, ions, 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 (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 2-body problem plus additional forces causing deviations from a hypothetical unperturbed 2-body trajectory.

Notes

  1. ^ "Historical Notes: Three-Body Problem". Retrieved December 2010. {{cite web}}: Check date values in: |accessdate= (help)
  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. ^ Restricted Three-Body Problem, Science World.
  5. ^ I. I. Gol’dman and V. D. Krivchenkov. Problems in Quantum Mechanics.. 3rd ed. Mineola, N.Y.: Dover Publications, 2006. 288 pp.

See also

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.
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