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''. See also Carter constant Hydrogen molecular ion Jacobi integral Lagrangian point Liouville dynamical system Three-body problem References Further reading External links The Euler Archive Orbits