A summary about a method for locking magnets in air

 The below plain text summarizes the underlying logic of magnetic bound state solutions covered in the article http://doi.org/10.3390/sym13030442 . It was originally posted in a forum and revised here.

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's theorem forbids this to happen in a static 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, that 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 (𝑟) 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/multipoles 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 which provide the attraction. The orbital mechanism is a dynamic interaction where field based forces are balanced by the inertial forces and Earnshaw's 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 (𝜋/2 < θ ≤ 𝜋 ) within a magnetic field. From basic experiences with magnets we know we cannot do it, by placing two magnets close to each other in a repulsive orientation unless restricting their motions. However this can happens 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. Here, a magnetic dipole body is exposed to a rotating magnetic field, causing it to perform an oscillatory motion. 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 an 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 regardless of its position and orientation. The video http://youtu.be/ZofshixkMg4 shows this effect at the time it was discovered. Please note there that the pendulum magnet bounces from the rotating field like it is bouncing from a rigid surface, pointing a sharp increase of the repulsion at proximity of the rotating magnet. 

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 polar orientation of the driving dipole making slightly of 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 a 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 also be primary depending on configurations. While the bounding mechanism is explained by behaviour of a free magnetic body interacting with a rotating dipole field, it is experimentally shown that a rotating magnetic dipole body having degrees of freedom can be bound to a static magnetic field (http://youtube.com/shorts/07Qhc4mB9Z4).

The channel http://youtube.com/user/sudanamaru covers numerous experimental solutions based on this principle which some of them are also mentioned in the article. Details about the experiment shown can be found in the description text.