Classification of the trajectories of uncharged particles in the Schwarzschild-Melvin metric

The paper and the candidate solution develop essentially the same reduction and bifurcation analysis for timelike geodesics in the Schwarzschild–Melvin (Ernst) spacetime: they derive the reduced Hamiltonian and effective potential U(r,θ), prove that any finite-radius relative equilibrium is equatorial, parametrize the circular equilibria via Σ(r) with Z(r)>0, identify the cusp via the same polynomial N(rc), obtain the far-field bifurcation curve Σ(∞), and use Σ(r)∪Σ(∞) to classify four Hill-region topologies. Minor discrepancies are: (i) the paper states the Hessian of U is negative definite on the unstable branch, which conflicts with the explicit expression ∂θθU=(4−B^2r_c^2)/(4Z)>0 along Σ(r) (from E real) and should instead be indefinite; the model’s stability classification matches the correct signs, (ii) the model briefly mislabels a denominator sign but reaches the correct conclusion sign(∂^2_rU)=sign N, and (iii) the model supplies a closed-form derivation of the threshold B_n and r_max(B) omitted in the paper. Overall conclusions and bifurcation structure coincide, including the cusp, the threshold B_n≈0.379, and the four-region partition (Σ(∞) range, etc.) .