McGehee blow-up of the Kepler problem on surfaces of constant curvature (Q2301727)

In this paper the authors address the question of the Kepler problem defined on a two-dimensional surface of nonzero constant curvature, namely the sphere (\(\mathbb{S}^2\)) and the hyperboloid (\(\mathbb{H}^2\)), into the Hamiltonian formalism. In particular, they are interested in the dynamics of this problem in a neighborhood of collisions. To obtain the desired result, the authors use McGehee transformation [\textit{R. McGehee}, Invent. Math. 27, 191--227 (1974; Zbl 0297.70011)] to blow-up the singularity that corresponds to the collision to a two-dimensional invariant manifold, called the collision manifold, which is boundary for all of the energy surfaces and not dependent on curvature. This allows the authors to read off the behavior of near-collision orbits. New results in this paper include a thorough description of the flow on the collision manifold, which is identical with the Newtonian Kepler problem. Unlike in the Newtonian case, however, the structure of the homothetic orbits are different. The calculations of the paper are performed with great detail.