It is shown that besides the known magnetic Lorentz, Thomas, and Kerst forces which act on a particle in a magnetic field with spatial variation, there is also a nonlinear sign-changing azimuthal force $F_{\phi} = —\frac{e}{c} [V_rH_s]$ caused by the nonlinear radial component of velocity and the S-th harmonic of the vertical component of the magnetic field, in the zone of total resonance in terms of the free radial fluctuations of the $S=Q_r$ index in perturbated orbits. This force leads to a post-resonance reduction in the amplitude of the forced radial oscillations (damping beatings) with a rise in the frequency $Q_r$ with preservation of the beam emittance.
Research Institute of Nuclear Physics, Moscow State University, Leninskie Gory, Moscow, 119992, Russia
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