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 . 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 moving in the gravitational field of two stars, 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 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.<ref name="lagrange" >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.</ref> 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. 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 other words, the total energy E is a constant of motion. The potential energy is given by 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