HYPERBOLIC MOTIONS IN THE N-BODY PROBLEM WITH HOMOGENEOUS POTENTIALS

The paper proves, for all 0<α<2, the existence of hyperbolic motions with prescribed asymptotic configuration v and positive energy h from any initial configuration x by constructing h–free-time minimizers and deriving sharp asymptotics via the Lagrange–Jacobi identity and quantitative estimates; the argument is complete and collision-free away from the endpoints. By contrast, the model’s solution hinges on a Busemann/weak KAM construction at positive energy that is only established in the literature for the Newtonian case (α=1) and, in the model, is asserted to extend “verbatim” to 0<α<2 without proof. Moreover, the model mixes incompatible sign conventions for the Lagrangian/Hamiltonian and the Hamilton–Jacobi PDE, so its calibration/energy identity does not rigorously match Newton’s equations as stated. Hence the paper’s proof is correct, while the model’s is not established as written.