Motions of a charged particle in the electromagnetic field induced by a non-stationary current.

The paper rigorously derives the electromagnetic potential via retarded potentials and reduces the Newton–Lorentz dynamics to a time-periodic 1-DOF radial equation with potential V(t,r)=L^2/(2r^2)+1/2[p_z+I0 ln r+k a(t,r)]^2, then proves (i) persistence of a T-periodic radial solution for non-resonant triples and (ii) twist-type stability, subharmonics, and (generalized) quasiperiodicity under a strong non-resonance, using Ortega’s third-approximation method and standard twist-map theory. The candidate solution replaces the correct potential by A_z(t,r)=−(I0+k I(t)) ln r (i.e., Biot–Savart with a time-varying coefficient), which is incompatible with time-dependent Maxwell equations and the paper’s distributional/retarded-potential construction; it also suppresses the essential r-dependence in a(t,r) and its radial derivative. While the candidate’s dynamical arguments (implicit function theorem; twist-map consequences) would go through for the correct potential after replacing I(t) ln r by a(t,r), the stated fields and the resulting radial equation are physically and mathematically incorrect for k≠0. The paper contains a minor slip in the statement/derivation of ż(t)≈I0 as k→0 (the correct limit is p_z+I0 ln r̄), but this does not affect the main results.