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). 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 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, it was found that the energy eigenvalue was mathematically a generalization of the Lambert W function (see Lambert W function 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. The fact that its solution is an implicit function 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