physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

and

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the

gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...

of two other point masses that are fixed in space. It is a particular version of the

three-body problem 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' ...

. This version of it is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate

spheroids A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...

. This problem is named after

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

, who discussed it in memoirs published in 1760. Important extensions and analyses to the three body problem were contributed subsequently by

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph Liouville 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 ...

Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...

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. Biography Jacobi was ...

Urbain Le Verrier Urbain Jean Joseph Le Verrier (; 11 March 1811 – 23 September 1877) was a French astronomer and mathematician who specialized in celestial mechanics and is best known for predicting the existence and position of Neptune using only mathematics. ...

William Rowan Hamilton Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

and

George David Birkhoff George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body ...

, among others. 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. 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 is also referred to as ℘-functions and they are usually denoted by the s ...

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 In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

and

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

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 vector (LRL vector) is a vector (geometric), vector used chiefly to describe the shape and orientation of the orbit (celestial mechanics), orbit of one astronomical body around another, such as a ...

as limiting cases. 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. \mathbf(\mathbf) = F( \mathbf ) where F is a force vector, ''F'' is a scalar valued force function (whose abso ...

s, such as the

electrostatic interaction Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), meani ...

described by

Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...

. The classical solutions of the Euler problem have been used to study

chemical bonding A chemical bond is the association of atoms or ions to form molecules, crystals, and other structures. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds or through the sharing of electrons as in ...

, 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 a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...

in 1921 in his doctoral dissertation under

Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld (; 5 December 1868 – 26 April 1951) was a German Theoretical physics, theoretical physicist who pioneered developments in Atomic physics, atomic and Quantum mechanics, quantum physics, and also educated and ...

, a study of the first ion of molecular hydrogen, namely the hydrogen molecular ion H₂⁺. These energy levels can be calculated with reasonable accuracy using the Einstein–Brillouin–Keller method, which is also the basis of the

Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model was a model of the atom that incorporated some early quantum concepts. Developed from 1911 to 1918 by Niels Bohr and building on Ernest Rutherford's nuclear Rutherford model, model, i ...

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 Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...

. 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 Darboux'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 the observed "intensity" of a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental ca ...

, such as

Newtonian gravity Newton's law of universal gravitation describes gravity as a force by stating 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 sq ...

. Examples of Euler's problem include an

electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

moving in the

electric field An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...

of two nuclei, such as the

hydrogen molecule-ion The dihydrogen cation or molecular hydrogen ion is a cation (positive ion) with formula H2^+. It consists of two hydrogen nuclei (protons), each sharing a single electron. It is the simplest molecular ion. The ion can be formed from the ioniz ...

. 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, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...

moving in the gravitational field of two

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

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.

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, Diarmuid Ó Mathúna published a book entitled "Integrable Systems in Celestial Mechanics". In this book, he 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 Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

; in other words, the total energy

E

is a

constant of motion In mechanics, a constant of motion is a physical quantity 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 tha ...

. The

potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...

is given by :

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

where

\mathbf

represents the particle's position, and

r_1

and

r_2

are the distances between the particle and the centers of force;

\mu_1

and

\mu_2

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 form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...

E = \frac + V(\mathbf)

where

m

and

\mathbf

are the particle's mass and

linear momentum In Newtonian mechanics, momentum (: momenta or momentums; 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. I ...

, 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 :

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

where

2\,a

is the separation of the two centers of force,

\theta_1

and

\theta_2

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 Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th century who contributed widely to applied mathemat ...

in his work on analytical mechanics,Whittaker

Analytical Dynamics of Particles and Rigid Bodies ''A Treatise on the Analytical Dynamics of Particles and Rigid Bodies'' is a treatise and textbook on analytical dynamics by British mathematician Sir Edmund Taylor Whittaker. Initially published in 1904 by the Cambridge University Press, the ...

, 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 = \mathbf^2 +  a^2 p_n^2 + 2\,a\,x_n \left(\frac - \frac \right),

where

p_n

denotes the momentum component along the

x_n

axis on which the attracting centers are located. This constant of motion corresponds to the total

squared

\mathbf^2

in the limit when the two centers of force converge to a single point (

a\rightarrow 0

), and proportional to the

\mathbf

in the limit when one of the centers goes to infinity (

a\rightarrow\infty

while

, x_n - 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 locat ...

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 de ...

. 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 func ...

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. 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 the product of a real number and 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 (algebra), square of an im ...

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 as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

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 gen ...

. 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

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

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

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 that varies in strength as the inverse square of the distance between them. The force may be either attra ...

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

\,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 In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point, as functions of one or several variables called parameters. In the case of a single parameter, parametric equations are commonly used to ...

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 i ...

Overview and history

Constants of motion

Quantum mechanical version

Generalizations

Mathematical solutions

Original Euler problem

See also

Notes

References

Further reading

External links