Projective Transformations for Regularized Central Force Dynamics: Hamiltonian Formulation

The paper derives a projective/canonical transformation (m = −1) and time reparametrizations dt = r^2 ds and dt = r^2 dτ/ℓ under which the Kepler–Manev Hamiltonian becomes linear in the s- or τ-parameters. In particular, with H(q,u,p,p_u) = (1/2) u^2 (ℓ^2 + u^2 p_u^2) − k1 u − (1/2) k2 u^2 (Eq. (189) in the paper), the s-dynamics satisfy q' = −ℓ★·q, p' = −ℓ★·p, u' = u^2 p_u, p_u' = −ℓ^2/u − 2u p_u^2 + k1/u^2 + k2/u, yielding linear subsystems with solutions identical to those presented by the model (see the paper’s Eqs. around the time reparametrizations and s-dynamics, and the linear u'' + (ℓ^2 − k2)u = k1 with ω^2 := ℓ^2 − k2; cf. the paper’s presentation of the Manev/Kepler cases and closed-form solutions). The model’s steps mirror the paper: conservation of q^2 and λ = (q·p)/q^2, linear angular motion on S^2, and the radial oscillator for u, as well as the τ-parameter form. No substantive discrepancies remain after aligning notation; hence both are correct and essentially the same derivation. Key correspondences: the Hamiltonian form and reparametrizations , the linear angular subsystem and constancy of ℓ in the central (unperturbed) case , and the Manev/Kepler radial oscillator and solutions .