On a linearly damped 2 body problem

The paper proves global non-vanishing and quantitative bounds for the damped inverse-square ODE by passing to polar form, deriving r^2 θ'(t)=M e^{-δ t}, and introducing a Lyapunov/energy functional F(t) that is strictly decreasing; from this it obtains r(t)≥η e^{-2δ t} and |r'(t)|≤C e^{δ t} (Theorem 2.1), then lifts the bounds to u with |u'(t)|≤D e^{δ t} (Corollary 2.2) . The model’s solution reproduces the same structural steps (polar reduction, angular momentum decay, energy dissipation) but derives the lower bound via a contradiction argument using s(t)=r(t)e^{2δ t}, and bounds |u'| using H(t)≤H(0)+c/r. Hence both are correct; the proofs are closely related but not identical.