I wrote a note about stability of magnetic bound state which evaluate the stability partially based on the zenithal angle of the free body.
Preprint: https://doi.org/10.31219/osf.io/uymc2
A note on the stability of the angular motion of a free body with a magnetic moment exposed to a rotating magnetic field
Hamdi Ucar
June 29, 2023
A basic magnetic bound state solution in classical physics requires the free body to perform a distinct cyclic angular motion as a result of its interaction with a rotating magnetic field. This motion resembles that of the arm of a spherical pendulum, tracing a conical pattern. The orientation of the pendulum arm corresponds to the orientation of the free body which is identified by its magnetic moment vector. According to the model in a basic configuration, this vector rotates synchronously with the rotating field on the same axis and maintains a constant zenithal angle, referred to as φ. While this angle satisfies the equilibrium between the magnetic torque exerted on the body and its inertial response, the stability analysis of this equilibrium with respect to the angle φ has not been fully explored mathematically. The complete analytical evaluation of the motion's stability can be challenging due to the non-linear nature of the equations of motion. However, we can examine a fundamental stability requirement to determine if the equilibrium might be stable. If this requirement is not met, the equilibrium is inherently unstable; otherwise, it may exhibit stability depending on the specific conditions provided. Through the evaluation of magnetic torque and angular displacement vectors it is found the dynamics satisfy a basic stability requirement.
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