A magnetic bound state in a general sense is keeping an entity or a formation localized by means of magnetic fields. In classical physics Earnshaw theorem forbids this to happen in a statistical configuration where no force fields including gravity, electrostatic, magnetic field and any combinations of them can be used to obtain a static equilibrium.
However, dynamic equilibriums are possible and this way one can obtain a bound state through orbital mechanisms like a star/planets system based on gravitational interaction and atoms where electrons orbit the nucleus based electric field attraction.
On the other hand, this mechanism does not work with magnetic fields since an orbital motion can be only stable in a central force problem when the dependence of the central force to the distance is expressed by a power factor greater than the inverse cube (r−3). This factor is r−2 for gravitational and electrical forces but r−4 for magnetic forces since magnetic force should always be between dipoles but electrical forces can be between charges (monopoles).
So, in classical physics, the only known way to obtain a bound state was the orbital mechanism and this only works with gravitational and electrical fields providing the attraction. The orbital motion is a dynamic interaction where field based forces are balanced by the inertial forces and Earnshaw theorem does not apply.
The magnetic bound state scheme given on this paper is also based on interplay of force fields and inertial forces, but rather in angular terms as angular motion, torque and moment of inertia. Since the bounded object is only required to perform an angular motion around its center of mass, it can stay without translational motion in contrast to the orbital motion.
Actually this angular motion only serves to obtain stability in angular degrees of freedom keeping a dipole body in an antiparallel orientation within a magnetic field.
From basic experiences with magnets we know we cannot do it when we place two magnets close to each other in a repulsive orientation on a low friction/frictionless surface. However this can happen here by the help of a property of the harmonic motion.
In a harmonic motion, the displacement and the acting force are always in the opposite directions. In a driven harmonic motion which this effect is based, a magnetic dipole body is exposed to a rotating magnetic field. This interaction is highly nonlinear and lies between driven harmonic motion and parametric excitation.
Anyway, a harmonic oscillator can be associated with a natural frequency and in a driven harmonic motion this determines the phase of the driven motion with respect to the periodic driving motion at a given frequency. By excluding possible damping factors, this phase is zero when driving frequency is below the natural frequency and shifts to 180° above it. This phase factor is called phase lag and in order to obtain the antiparallel kind alignment mentioned above, phase lag should be 180°.
This way, it is possible to exert a force of a magnetic body having full degrees of freedom in the direction of the weak field. That is, a rotating dipole magnet can repel another dipole magnet having degrees of freedom. After this stage, it is possible to obtain a stable equilibrium between this repulsive interaction and an attractive field force allowing to establish a magnetic bound state.
It should be noted that the power factor of the repulsive interaction with respect to the distance between dipoles varies between r−7 to r−8, almost twice of the static forces between dipoles. This high power factor ensures the stability of the equilibrium where the interaction can be attractive at long distance and switch to repulsive when magnets get close.
We can also align the orientation of the driving dipole slightly off the rotation plane in order to create a virtual static dipole which can be used for the attraction factor. Therefore a dipole magnet attached to a rotor can lock another dipole magnet in air. While this interaction is explained by the angular oscillation of the free body, translational oscillations also present to some extent and contribute to the repulsive factor. This contribution can be also primary depending to configurations.
The channel [YouTube channel advertisement redacted by the Mentors] covers numerous experimental solutions on this principle which some also mentioned in the article. Details about shown experiments can be found in their description text.
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