COLLISIONAL DYNAMICS FOR STRONG-WEAK POTENTIAL HILL'S LUNAR PROBLEM

We study a particular dynamical system in terms of global existence and singularity. This model has various physical applications. For instance, it can be derived from the quasi -homogeneous three -body system in a rotating coordinate system with angular speed omega. On the other hand, quasi -homogeneous (U(r) = - A/r(a) - B/r(b) , where r is the mutual distance and A, B, a, b are positive constants) potential itself is of great interest. Important examples of quasi -homogeneous potentials include the Schwarzschild potential (U Schwarzschild (r) = - A/r - B/r(3) ) and the Manev potential (U-Manev(r) = -A/r - B/r(2)). This paper is partitioned into two major parts. In the first part, we classify the fate of a given initial condition dynamically under the flow of the dynamical system. We fully characterize the phase space under some energy threshold using an indicator function. This energy threshold is characterized variationally. In particular, for strong -weak potentials (b > 2 > a), our results demonstrate the existence of the "black hole effect" for omega sufficiently large: Collision sets, with non -zero Lebesgue measure, consisting of initial conditions that lead to finite -time collision, are constructed under and at an energy threshold. In the second part, we introduce the McGehee coordinate system, as well as a new time scale, to study near -collision dynamics of the system. By applying the McGehee transformation, we are able to derive the asymptotic configuration with respect to time for any collision solution at collision.