A study of the plane unrestricted three-body problem (Q1370941)

From the summary: The general (unrestricted) three-body problem is investigated in the case when the force of mutual attraction between the bodies is proportional to the \(n\)th power of their distance, where \(n\) is an arbitrary real number. A new description is given of the plane problem, based on the introduction of the following Lagrange variables: \(r\) -- the square root of half the polar moment of inertia, \(\psi\) -- the angle between the two sides of the triangle, and \(y\) -- the natural logarithm of the quotient of those two sides. Routh's equations are derived, in which the variable \(r\) is `almost separated' from \(y\) and \(\psi\); the system of equations is reversible. It is shown that the qualitative results, known for Newtonian interaction \((n=-2)\), are valid throughout the range \(-3<n <-1\). In particular, for these values of \(n\) `elementary' methods of analysis are used to solve the problems of Hill stability for a pair of bodies. The existence of final motions relating to hyperbolic-elliptic motions is established for \(n=-2\), and a local analysis is carried out of the neighbourhoods of the classical libration points.