Euler's problem also covers the case when the particle is acted upon by other inverse-square central forces, such as the electrostatic interaction described by Coulomb's law. 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 in his doctoral dissertation under Arnold Sommerfeld, a study of the first ion of molecular hydrogen, 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 of atomic hydrogen.
By treating Euler's problem as a Liouville dynamical system, the exact solution can be expressed in terms of elliptic integrals. 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 and angular momentum are not, since the two fixed centers can apply a net force and torque. Neverthless, the particle has a second conserved quantity that corresponds to the angular momentum or to the Laplace-Runge-Lenz vector as limiting cases.
The Euler three-body problem is known by a variety of names, such as the the problem of two fixed centers, the Euler-Jacobi problem, and the two-center Kepler problem. It is also called the circular restricted three-body problem, as it is a special case of the general three-body problem in which the two massive particles have zero orbital eccentricity and the third particle is "restricted" to have negligible mass. Special cases of this problem include the Copenhagen problem (where the two large masses are equal) and the Pythagorean problem (where the three bodies have initial masses in the ratio 3:4:5 and prescribed initial positions). 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 and Velde's problem.
Euler's three-body problem is to describe the motion of a particle under the influence of two centers that attract the particle with central forces that decrease with distance as an inverse-square law, such as Newtonian gravity or Coulomb's law. Examples of Euler's problem include a planet moving in the gravitational field of two stars, or an electron moving in the electric field of two nuclei, such as the first ion of the hydrogen molecule H2+. The strength of the two inverse-square forces need not be equal; for illustration, the two attracting stars may have different masses, and the two nuclei may have different charges, as in the molecular ion HeH2+.
This problem was first considered by Leonhard Euler, who showed that it had an exact solution in 1760. Joseph Louis Lagrange solved a generalized problem in which the centers exert both linear and inverse-square forces. Carl Gustav Jacob Jacobi 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.
where r represents the particle's position, and r1 and r2 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
where m and p are the particle's mass and linear momentum, respectively.
The particle's linear and angular momentum 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
where 2a 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,, and generalized to n dimensions by Coulson and Joseph in 1967. In the Coulson-Joseph form, the constant of motion is written
This constant of motion corresponds to the total angular momentum |L|2 in the limit when the two centers of force converge to a single point (a→0), and proportional to the Laplace-Runge-Lenz vector A in the limit when one of the centers goes to infinity (a→∞ while x-a remains finite).
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 augment the inverse-square force laws with a force that increases linearly with distance. The next generalization is to add a third center of force midway the original two centers, that exerts only a linear force. The final set of generalizations is to add two fixed centers of force at positions that are imaginary numbers, with forces that are both linear and inverse-square laws, 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 cigar. 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 imaginary numbers. The oblate spheroid solution is astronomically more important, since most planets, stars and galaxies are approximately oblate spheroids; prolate spheroids are very rare.
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
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. Therefore, according to Bonnet's theorem, the same ellipses are the solutions for the Euler problem.
Introducing elliptic coordinates,
the potential energy can be written as
and the kinetic energy as
This is a Liouville dynamical system if ξ and η are taken as φ1 and φ2, respectively; thus, the function Y equals
and the function W equals
Using the general solution for a Liouville dynamical system, one obtains
Introducing a parameter u by the formula
gives the parametric solution
Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.