physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

and

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, 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 fixed in space. This problem is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids. This problem is named after

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

, who discussed it in memoirs published in 1760. Important extensions and analyses were contributed subsequently by

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLiouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...

Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summariz ...

, Jacobi,

Darboux Darboux is a surname. Notable people with the surname include: *Jean Gaston Darboux (1842–1917), French mathematician *Lauriane Doumbouya (née Darboux), the current First Lady of Guinea since 5 September 2021 *Paul Darboux Paul Darboux (May 10, ...

, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff and E. T. Whittaker, among others. Euler's problem also covers the case when the particle is acted upon by other inverse-square

central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...

s, such as the electrostatic interaction described by

Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...

. The classical solutions of the Euler problem have been used to study chemical bonding, using a semiclassical approximation of the energy levels of a single electron moving in the field of two atomic nuclei, such as the diatomic ion HeH²⁺. This was first done by

Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...

in his doctoral dissertation under

Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretic ...

, a study of the first ion of molecular hydrogen, namely the hydrogen molecule-ion H₂⁺. These energy levels can be calculated with reasonable accuracy using the

Einstein–Brillouin–Keller method The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from B ...

, which is also the basis of the

Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Sy ...

of atomic hydrogen. More recently, as explained further in the quantum-mechanical version, analytical solutions to the eigenvalues (energies) have been obtained: these are a ''generalization'' of the

Lambert W function In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse relation of the function , where is any complex number and is the exponential func ...

. The exact solution, in the full three dimensional case, can be expressed in terms of

Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...

For convenience, the problem may also be solved by numerical methods, such as Runge–Kutta integration of the equations of motion. The total energy of the moving particle is conserved, but its

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

and

angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed sy ...

are not, since the two fixed centers can apply a net force and torque. Nevertheless, the particle has a second conserved quantity that corresponds to the

or to the

Laplace–Runge–Lenz vector In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For t ...

as limiting cases. The Euler three-body problem is known by a variety of names, such as the problem of two fixed centers, the Euler–Jacobi problem, and the two-center Kepler problem. Various generalizations of Euler's problem are known; these generalizations add linear and inverse cubic forces and up to five centers of force. Special cases of these generalized problems include ''

's problem'' Darboux JG, ''Archives Néerlandaises des Sciences'' (ser. 2), 6, 371–376 and ''Velde's problem''.Velde (1889) ''Programm der ersten Höheren Bürgerschule zu Berlin''

Overview and history

Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with

s that decrease with distance as an

inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unde ...

, such as

Newtonian gravity Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

. Examples of Euler's problem include an

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...

moving in the electric field of two nuclei, such as the hydrogen molecule-ion . The strength of the two inverse-square forces need not be equal; for illustration, the two nuclei may have different charges, as in the molecular ion HeH²⁺. In Euler's three-body problem we assume that the two centres of attraction are stationary. This is not strictly true in a case like , but the protons experience much less acceleration than the electron. However, the Euler three-body problem does not apply to a

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...

moving in the gravitational field of two

star A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth make ...

s, because in that case at least one of the stars experiences acceleration similar to that experienced by the planet. This problem was first considered by

, who showed that it had an exact solution in 1760. Euler L, ''Nov. Comm. Acad. Imp. Petropolitanae'', 10, pp. 207–242, 11, pp. 152–184; ''Mémoires de l'Acad. de Berlin'', 11, 228–249.

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia Lagrange JL, ''Miscellanea Taurinensia, 4, 118–243; ''Oeuvres'', 2, pp. 67–121; ''Mécanique Analytique'', 1st edition, pp. 262–286; 2nd edition, 2, pp. 108–121; ''Oeuvres'', 12, pp. 101–114.

Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasio ...

showed that the rotation of the particle about the axis of the two fixed centers could be separated out, reducing the general three-dimensional problem to the planar problem. Jacobi CGJ, ''Vorlesungen ueber Dynamik'', no. 29. ''Werke'', Supplement, pp. 221–231 In 2008, Birkhauser published a book entitled "Integrable Systems in Celestial Mechanics". In this book an Irish mathematician, Diarmuid Ó Mathúna, gives closed form solutions for both the planar two fixed centers problem and the three dimensional problem.

Constants of motion

The problem of two fixed centers conserves

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

; in other words, the total energy ''E'' is a

constant of motion In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather than ...

. The

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

is given by :

V(\mathbf) = - \frac - \frac

where r represents the particle's position, and ''r''₁ and ''r''₂ are the distances between the particle and the centers of force; μ₁ and μ₂ are constants that measure the strength of the first and second forces, respectively. The total energy equals sum of this potential energy with the particle's

kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its a ...

E = \frac \left,  \mathbf \^2 + V(\mathbf)

where ''m'' and p are the particle's mass and

linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

, respectively. The particle's

and

are not conserved in Euler's problem, since the two centers of force act like external forces upon the particle, which may yield a net force and torque on the particle. Nevertheless, Euler's problem has a second constant of motion :

r_^ r_^ \left( \frac \right) \left( \frac \right) - 
2a \left \mu_ \cos \theta_ + \mu_ \cos \theta_ \right

where 2''a'' is the separation of the two centers of force, θ₁ and θ₂ are the angles of the lines connecting the particle to the centers of force, with respect to the line connecting the centers. This second constant of motion was identified by E. T. Whittaker in his work on analytical mechanics,Whittaker Analytical Dynamics of Particles and Rigid Bodies, p. 283. and generalized to ''n'' dimensions by Coulson and Joseph in 1967. In the Coulson–Joseph form, the constant of motion is written :

B = \left,  \mathbf \^2 +  a^2 \left,  \mathbf \^2 
-2a \left mu_ \cos \theta_1 + \mu_2 \cos \theta_2 \right

This constant of motion corresponds to the total

, L, ² in the limit when the two centers of force converge to a single point (''a'' → 0), and proportional to the

A in the limit when one of the centers goes to infinity (''a'' → ∞ while ''x'' − ''a'' remains finite).

Quantum mechanical version

A special case of the quantum mechanical three-body problem is the hydrogen molecule ion, . Two of the three bodies are nuclei and the third is a fast moving electron. The two nuclei are 1800 times heavier than the electron and thus modeled as fixed centers. It is well known that the Schrödinger wave equation is separable in

prolate spheroidal coordinates Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are locate ...

and can be decoupled into two ordinary differential equations coupled by the energy eigenvalue and a separation constant. However, solutions required series expansions from basis sets. Nonetheless, through

experimental mathematics Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...

, it was found that the energy eigenvalue was mathematically a ''generalization'' of the Lambert W function (see

and references therein for more details). The hydrogen molecular ion in the case of clamped nuclei can be completely worked out within a

Computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The ...

. The fact that its solution is an

implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit fun ...

is revealing in itself. One of the successes of theoretical physics is not simply a matter that it is amenable to a mathematical treatment but that the algebraic equations involved can be symbolically manipulated until an analytical solution, preferably a closed form solution, is isolated. This type of solution for a special case of the three-body problem shows us the possibilities of what is possible as an analytical solution for the quantum three-body and many-body problem.

Generalizations

An exhaustive analysis of the soluble generalizations of Euler's three-body problem was carried out by Adam Hiltebeitel in 1911. The simplest generalization of Euler's three-body problem is to add a third center of force midway between the original two centers, that exerts only a linear Hooke force (confer

Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...

). The next generalization is to augment the inverse-square force laws with a force that increases linearly with distance. The final set of generalizations is to add two fixed centers of force at positions that are

imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...

s, with forces that are both linear and

s, together with a force parallel to the axis of imaginary centers and varying as the inverse cube of the distance to that axis. The solution to the original Euler problem is an approximate solution for the motion of a particle in the gravitational field of a prolate body, i.e., a sphere that has been elongated in one direction, such as a cigar shape. The corresponding approximate solution for a particle moving in the field of an oblate spheroid (a sphere squashed in one direction) is obtained by making the positions of the two centers of force into

s. The oblate spheroid solution is astronomically more important, since most planets, stars and galaxies are approximately oblate spheroids; prolate spheroids are very rare. The analogue of the oblate case in

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

is a

Kerr black hole The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of ...

. The geodesics around this object are known to be integrable, owing to the existence of a fourth constant of motion (in addition to energy, angular momentum, and the magnitude of four-momentum), known as the Carter constant. Euler's oblate three body problem and a Kerr black hole share the same mass moments, and this is most apparent if the metric for the latter is written in Kerr–Schild coordinates. The analogue of the oblate case augmented with a linear Hooke term is a Kerr–de Sitter black hole. As in

, the

cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...

term depends linearly on distance from the origin, and the Kerr–de Sitter spacetime also admits a Carter-type constant quadratic in the momenta.Charalampos Markakis, Constants of motion in stationary axisymmetric gravitational fields, MNRAS (July 11, 2014) 441 (4): 2974-2985. doi: 10.1093/mnras/stu715, https://arxiv.org/abs/1202.5228

Mathematical solutions

Original Euler problem

In the original Euler problem, the two centers of force acting on the particle are assumed to be fixed in space; let these centers be located along the ''x''-axis at ±''a''. The particle is likewise assumed to be confined to a fixed plane containing the two centers of force. The potential energy of the particle in the field of these centers is given by :

V(x, y) = \frac - \frac .

where the proportionality constants μ₁ and μ₂ may be positive or negative. The two centers of attraction can be considered as the foci of a set of ellipses. If either center were absent, the particle would move on one of these ellipses, as a solution of the

Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ...

. Therefore, according to Bonnet's theorem, the same ellipses are the solutions for the Euler problem. Introducing

elliptic coordinates In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively ...

, :

\,x = \,a \cosh \xi \cos \eta,

\,y = \,a \sinh \xi \sin \eta,

the potential energy can be written as :

& = \frac, \end

and the kinetic energy as :

T = \frac \left( \cosh^ \xi - \cos^ \eta \right) \left( \dot^ + \dot^ \right).

This is a Liouville dynamical system if ξ and η are taken as φ₁ and φ₂, respectively; thus, the function ''Y'' equals :

\,Y = \cosh^ \xi - \cos^ \eta

and the function ''W'' equals :

W = -\mu_ \left( \cosh \xi + \cos \eta \right) - \mu_ \left( \cosh \xi - \cos \eta \right).

Using the general solution for a Liouville dynamical system, one obtains :

\frac \left( \cosh^ \xi - \cos^ \eta \right)^ \dot^ = E \cosh^ \xi + \left( \frac \right) \cosh \xi - \gamma

\frac \left( \cosh^ \xi - \cos^ \eta \right)^ \dot^ = -E \cos^ \eta + \left( \frac \right) \cos \eta + \gamma

Introducing a parameter ''u'' by the formula :

du = \frac = 
\frac,

gives the parametric solution :

u = \int \frac = 
\int \frac.

Since these are

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...

s, the coordinates ξ and η can be expressed as elliptic functions of ''u''.

References

External links

The Euler Archive
{{DEFAULTSORT:Euler's Three-Body Problem Orbits