In 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 fixed in space.
This version of it is exactly solvable, and yields an approximate solution for particles moving in the gravitational fields of prolate and oblate spheroids.
Important extensions and analyses to the three body problem were contributed subsequently by Joseph-Louis Lagrange, Joseph Liouville, Pierre-Simon Laplace, Carl Gustav Jacob Jacobi, Urbain Le Verrier, William Rowan Hamilton, Henri Poincaré and George David Birkhoff, among others.
The exact solution, in the full three dimensional case, can be expressed in terms of Weierstrass's elliptic functions[2] 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.
Nevertheless, the particle has a second conserved quantity that corresponds to the angular momentum or to the Laplace–Runge–Lenz vector as limiting cases.
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+.
[3] 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.
[4][5] 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.
[7] 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 an electron moving in the electric field of two nuclei, such as the hydrogen molecule-ion H+2.
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+.
[8] Joseph Louis Lagrange solved a generalized problem in which the centers exert both linear and inverse-square forces.
[10] In 2008, Diarmuid Ó Mathúna published a book entitled "Integrable Systems in Celestial Mechanics".
A special case of the quantum mechanical three-body problem is the hydrogen molecule ion, H+2.
The hydrogen molecular ion in the case of clamped nuclei can be completely worked out within a Computer algebra system.
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.
An exhaustive analysis of the soluble generalizations of Euler's three-body problem was carried out by Adam Hiltebeitel in 1911.
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.
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.
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.
If either center were absent, the particle would move on one of these ellipses, as a solution of the Kepler 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,[17] 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.