Three-body problem
Encyclopedia
Three-body problem has two distinguishable meanings in physics
and classical mechanics
:
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun.
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.
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.
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.
, various atomic systems can be described as 3-body systems: Atoms of helium
or the helium-like ions for example; however, at high atomic number
s, 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.
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.
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.
In the 21st century, experiments with atomic traps and molecular traps enhance the possibilities to deal with 3-body systems.
Upon excitation with short pulses, during the short time after the excitation, such systems may show trajectories and other attributes typical of classical mechanics
.
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
.
s around their common center of mass
, and the third mass is small and moves in the same plane. 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
.
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
;
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
. As for classical mechanics
, the variety of divergent trajectories with various Lyapunov exponent
s 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?
In the case of quantum mechanics, the main part of the 3-body problem refers to the finding the eigenstates and their energies.. 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 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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
:
- 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 motionNewton's laws of motionNewton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...
and of universal gravitation. - In an extended modern sense a three-body problem is a class of problems in classicalClassical mechanicsIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
or quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
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 potentialScalar potentialA scalar potential is a fundamental concept in vector analysis and physics . The scalar potential is an example of a scalar field...
such as gravityGravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
or electromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
.
Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth and the Sun.
History
The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
published his 'Principia' (Philosophiae Naturalis Principia Mathematica
Philosophiae Naturalis Principia Mathematica
Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...
). 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
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...
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.
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.
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 lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
c.
In quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, at high speed, the creation
Matter creation
Matter creation is the process inverse to particle annihilation. It is the conversion of massless particles into one or more massive particles. This process is the time reversal of annihilation. Since all known massless particles are bosons and the most familiar massive particles are fermions,...
and annihilation
Annihilation
Annihilation is defined as "total destruction" or "complete obliteration" of an object; having its root in the Latin nihil . A literal translation is "to make into nothing"....
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 constantFine-structure constant
In physics, the fine-structure constant is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has constant numerical value in all systems of units...
, various atomic systems can be described as 3-body systems: Atoms of helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...
or the helium-like ions for example; however, at high atomic number
Atomic number
In chemistry and physics, the atomic number is the number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus. It is conventionally represented by the symbol Z. The atomic number uniquely identifies a chemical element...
s, the velocities become relativistic
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
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
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
, mass of the electron, and the Coulomb interaction between them.
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
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
, or the carbon dioxide
Carbon dioxide
Carbon dioxide is a naturally occurring chemical compound composed of two oxygen atoms covalently bonded to a single carbon atom...
) can be treated as a 3-body problem.
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.
In the 21st century, experiments with atomic traps and molecular traps enhance the possibilities to deal with 3-body systems.
Upon excitation with short pulses, during the short time after the excitation, such systems may show trajectories and other attributes typical of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
.
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
Factorization
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
, 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
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
.
Circular restricted three-body problem
In the circular restricted three-body problem two massive bodies move in circular orbitOrbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s around their common center of mass
Center of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
, and the third mass is small and moves in the same plane. With respect to a rotating reference frame
Rotating reference frame
A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth. A rotating frame of reference is a special case of a non-inertial reference...
, 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
Horseshoe orbit
A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body . The orbital period of the smaller body is very nearly the same as for the larger body, and its path appears to have a horseshoe shape in a rotating reference frame as viewed from the...
. It can be useful to consider the effective potential
Effective potential
The effective potential, or effective potential energy is a mathematical expression combining the centrifugal potential energy with the potential energy of a dynamical system. Commonly used in calculating the orbits of planets , the effective potential allows one to reduce a problem to fewer...
.
Classical versus quantum mechanics
Physicist Vladimir KrivchenkovVladimir Krivchenkov
Vladimir Dmitrievich Krivchenkov Russian physicist, professor, creator of the "Deterministic concept of Quantum Mechanics", author of textbooks, translated into many languages and published many times.....
used the 3-body problem as an example, showing the simplicity of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in comparison to classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. The quantum 3-body problem is studied in university courses of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
;
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
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. As for classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the variety of divergent trajectories with various Lyapunov exponent
Lyapunov exponent
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...
s 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?
In the case of quantum mechanics, the main part of the 3-body problem refers to the finding the eigenstates and their energies.. 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 problemN-body problem
The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
. 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
Qiudong Wang
Qiudong Wang is an Associate Professor at the Department of Mathematics, the University of Arizona. In 1982 he received a B.S. at Nanjing University and in 1994 a Ph.D. at the University of Cincinnati....
for n > 3, see n-body problem
N-body problem
The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
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
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
in the form of numerical integration
Numerical integration
In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...
(or, for some cases, classical trigonometric series approximations) is unavoidable, see n-body simulation
N-body simulation
An N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...
. 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
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
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
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
theory, in which the system is considered as a 2-body problem plus additional forces causing deviations from a hypothetical unperturbed 2-body trajectory.
See also
- Euler's three-body problemEuler's three-body problemIn physics and astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the gravitational field of two other point masses that are either fixed in space or move in circular coplanar orbits about their center of mass...
- Few-body systemsFew-body systemsIn quantum mechanics and classical mechanics, a few-body system consists of a small number of well-defined structures or point particles.In quantum mechanics, examples of few-body systems include light nuclear systems , small molecules, light atoms , atomic collisions, and quantum...
- n-body problemN-body problemThe n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...
- n-body simulationN-body simulationAn N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity . In cosmology, they are used to study processes of non-linear structure formation such as the process of forming galaxy filaments and galaxy halos from dark...
- Galaxy formation and evolutionGalaxy formation and evolutionThe study of galaxy formation and evolution is concerned with the processes that formed a heterogeneous universe from a homogeneous beginning, the formation of the first galaxies, the way galaxies change over time, and the processes that have generated the variety of structures observed in nearby...
- Numerical methods
- Two-dimensional gasTwo-dimensional gasA two-dimensional gas is a collection of N objects which are constrained to move in a planar or other two-dimensional space in a gaseous state. The objects can be: ideal gas elements such as rigid disks undergoing elastic collisions; elementary particles, or any object in physics which obeys laws...
- Hydrogen Molecular ion