As described below, the variables have been changed from qs to φs, and the functions us and ws substituted by their counterparts χs and ωs.
This solution has numerous applications, such as the orbit of a small planet about two fixed stars under the influence of Newtonian gravity.
In classical mechanics, Euler's three-body problem describes the motion of a particle in a plane under the influence of two fixed centers, each of which attract the particle with an inverse-square force such as Newtonian gravity or Coulomb's law.
The potential energy of the moving particle is given by The two centers of attraction can be considered as the foci of a set of ellipses.
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 below, 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.
, re-arranging, and exploiting the relation 2T = YF yields the equation which may be written as where E = T + V is the (conserved) total energy.