Euler's three-body problem

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. It is a particular version of the three-body problem. This version of it 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, 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 Liouville, Pierre-Simon Laplace, Carl Gustav Jacob Jacobi, Urbain Le Verrier, William Rowan Hamilton, Henri Poincaré and George David Birkhoff, 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 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.