Gyromotion

Gyromotion refers to the rapid circular motion of a charged particle around magnetic field lines caused by the Lorentz force. In a uniform magnetic field, the component of velocity perpendicular to the field produces circular motion with a gyroradius (Larmor radius), while the component parallel to the field remains unchanged, giving a helical trajectory along the field line. The frequency of this circular motion, the cyclotron or gyrofrequency, is ωc = |q| B / m in the nonrelativistic limit. The gyroradius is r_g = m v_perp / (|q| B). In relativistic contexts, ωc = |q| B / (γ m) and r_g = γ m v_perp / (|q| B), where γ is the Lorentz factor.

Gyromotion occurs on a fast timescale compared with slower variations in fields, which leads to the common guiding-center approximation in plasma physics. In this view, the instantaneous center of the circular orbit—the guiding center—drifts slowly under the influence of electric fields (E × B drift) and gradients or curvature of the magnetic field. The magnetic moment μ = m v_perp^2 /(2 B) is an adiabatic invariant when the magnetic field changes gradually, linking microscopic gyration to macroscopic plasma behavior.

Applications and relevance: gyromotion is fundamental in magnetized plasmas, including fusion devices, space and astrophysical plasmas, and beam physics. It governs confinement, pitch-angle scattering, wave-particle interactions, and energy transport. Realistic systems often involve nonuniform fields, electric fields, or relativistic effects that modify or couple gyromotion to the broader motion of the particle.