GLOBAL EXISTENCE AND SINGULARITY OF THE HILL’S TYPE LUNAR PROBLEM WITH STRONG POTENTIAL

The uploaded paper proves the sub-threshold dichotomy, threshold classification, and the near-threshold ejection/no-return/collision result for the planar Hill’s type lunar problem with homogeneous potential, defining W, the ground states ±Q, and the ground state energy E* as the minimum of E under W=0 (Lemma 1, Definition 2) ; it establishes invariance of W± below E*, globality in W+, bounded-or-collision in W−, and for α ≥ 2 finite-time collision in W− (Theorem 1, Lemma 6, Proposition 2) ; it also gives the threshold dynamics (Theorem 2, Proposition 3–4) and the near-threshold ejection/one-pass and collision for exits with W<0 (Theorem 3; Section 5) . The candidate solution reproduces these results with essentially the same virial/variational mechanism and invariant manifold arguments, differing only in notational conventions (e.g., using I = r^2 so I¨ = 4K + 4L + 2W versus the paper’s I = 1/2 r^2 so I¨ = K) and in minor exposition. No substantive logical discrepancies were found; the model correctly restricts the finite-time collision and one-pass collision criterion to the strong-force regime α ≥ 2 and does not overclaim on the W>0 “no-return” case that the paper leaves as a conjecture .