Linear and Regular Kepler-Manev Dynamics via Projective Transformations: A Geometric Perspective

The paper explicitly derives the projectively transformed Hamiltonian H = (1/2m)(r_n^2 ℓ^2 + π̃_n^2) + U_0(r_n) and, on each energy shell with Sundman scaling f = r_n^{-2}, obtains linear s-time dynamics: r_i'' + (ℓ/m)^2 r_i = 0 for the in-plane coordinates and r_n'' + (ℓ/m)^2 r_n = k_1/m when U_0(r_n) = -k_1 r_n - (1/2)k_2 r_n^2, equivalently r_n'' + β^2 r_n = k_1/m with β^2 := (ℓ/m)^2 - k_2/m. After the true-anomaly rescaling f = ( ℓ r_n^2 )^{-1}, the paper’s equation becomes r_{n,ττ} + (β^2/ℓ^2) r_n = k_1/ℓ^2. These statements and formulas appear explicitly in the paper’s summary of unperturbed dynamics and ODEs (Eq. (284)–(285)) and in the discussion of the transformed potential (Eq. (283)) . By contrast, the candidate solution asserts r_n'' + β^2 r_n = -k_1 (and, after normalization, r_{n,ττ} + (β^2/ℓ^2) r_n = -k_1/ℓ^2), which flips the sign of the constant forcing term; this contradicts the paper and the classical Binet equation u'' + u = µ/ℓ^2. Moreover, the candidate’s Jacobi Hamiltonian S := r_n^2(H - E) is said to have kinetic part (1/2m)(ℓ^2 + r_n^6 π_n^2). The paper’s formula implies instead S_{kin} = (1/2m)(r_n^4 ℓ^2 + r_n^6 π_n^2) (using π̃_n = r_n^2 π_n), i.e., there is a missing r_n^4 factor multiplying ℓ^2 in the model’s expression. The paper’s use of the conformally scaled vector field f X_H to derive first-order linear ODEs (hence constant-coefficient second-order ODEs) avoids these mistakes and is consistent with the invariant integrals ℓ_{ij} and ℓ^2 being conserved under Tψ . Therefore the paper’s argument is correct; the model’s solution contains a sign error in the normal equation and an incorrect kinetic splitting in S.