The problem of infinite spin for parabolic and collision solutions in the planar n-body problem

The paper proves “no infinite spin” for k-parabolic solutions (Theorem 1.8) and k-collision solutions (Theorem 1.14) under the assumption that the reduced, normalized shape converges to an isolated central configuration, using a modified McGehee transformation (2.8), the Euler–Lagrange system (2.1) with µ = r^2 θ̇ + r^2 Ω/||(s,1)||^2, the bound |θ̇| ≤ |µ|/r^2 + C||ω|| (2.6), a nonautonomous center manifold reduction (Theorem 1.13), and a Lojasiewicz-type gradient argument to prove finite Fubini–Study arclength, hence finite spin . In the collision case, they show µ(t) = O(r^{5/2}) from ∂θUk,k′ = O(r) and Sundman’s estimates, making µ/r^2 integrable . The candidate solution instead fixes the horizontal (mechanical-connection) gauge to set Ω(s,ω)=0, obtaining θ̇ = L_k/r^2, identifies L̇_k as the external torque, and shows L̇_k = O(t^{-4/3}) in the parabolic regime and L̇_k = O(r) with L_k → 0 in the collision regime; using r^2 ~ t^{4/3} and |ṙ| ≳ r^{-1/2} makes θ̇ integrable, so there is no infinite spin. These steps agree with the paper’s torque/decay scalings—e.g., ∂θUk,k′ = O(t^{-4/3}) in parabolic and O(r) near collision—and with the same isolated-CC hypothesis (Proposition 1.6 and Theorems 1.8, 1.14) . The model omits the paper’s center-manifold/Lojasiewicz machinery by working in a horizontal gauge; this is a different, valid route near the limiting isolated CC, though some energy-inequality details could be tightened.