REGULARIZING RECTIFICATIONS OF THE TWO-BODY PROBLEM

This paper is concerned with several rectifications (invariant transformations) involving two solutions of the two-body problem of any eccentricity. Such a transform, which can change an unbounded orbit into a bounded orbit has been described in [5]. Each having no zero angular momentum. This may be a regularization for infinity. Here, it is extended to the circular case. As it concerns only non rectilinear solution, it is not a regularization of the collision. For that purpose, another family of rectifications is considered. By involving an orthogonal affinity these transformations may regularise the collision but not infinity. By the product of these two sorts of rectifications, whose analytical expressions depend on the values of the eccentricities with respect to unity, in the two cases the problem becomes regular. Some of these transformations seem to be associated with the Laplace integral and for the n body problem, if they exist, perhaps with an integral invariant of order two, by way of the invariable plane of Laplace.