Relativistic Effects in the Dynamics of a Particle in a Coulomb Field

The paper proves that in special-relativistic central-force dynamics with real-analytic V and V′>0, no potential has the Bertrand property (periodicity of all trajectories near every circular orbit). This is stated as Theorem 2.1 and developed in Section 3 via a reduction to a relativistic oscillator for ρ=1/r and an isochrony argument that yields an overdetermined set of analytic constraints on V, culminating in a contradiction except for a degenerate case that still fails to give closed noncircular orbits . The candidate model arrives at the same conclusion by a different local argument: linearizing about a circular orbit to obtain the radial epicyclic frequency and then proving that the exact apsidal angle Δθ(E,ℓ) is strictly decreasing in the energy E near the circular energy (for fixed ℓ), which precludes periodicity for all nearby initial conditions. The two approaches agree on the main claim but use distinct techniques (period-function/isochrony vs. apsidal-angle monotonicity), so the appropriate verdict is that both are correct, with different proofs.