## Levitation... it is quite simple.

----This is the way the science is. The first reaction to something new - "it's impossible!" Some times later - "hm-m..., there is something in this". And at last - "oh, it's trivial, it's ... quite simple!!! ".   ----The conviction, that the levitation is impossible, is usually based on the strict mathematical statement of Brownbeck that it is impossible to build up a stable configuration of non-conducting charges (Irnshow theorem) or magnetic bodies placed in the medium with relative electric or magnetic inductivities more than or equal to 1. This statement is the consequence of the fact that the potential energy function of the body placed in a static potential (electrical, magnetic, gravitational...) in the absence of field sources around the body may have the only type of singular points - saddle points. That is no potential wells can exist in such a system, where the body could stay for a long time. However, what is impossible in the statics may be possible in the dynamics (e.g. when the body is in the variable external field or when many bodies are moving in the fields of each other or the body is not point).

The possibility of the dynamical stability one could first see in the work of Mathieu (1838) devoted to the membrane vibration problem. It appears that Mathieu equation:

##### dY/dZ +(h+2e*cos(2*Z))*X=0 (1) .

where X = dX / dZ , Z = w * t, w is frequency of vibration , allows the dynamical stability of the state, which is unstable in the statics (at h <0) beyond the resonance conditions. An example of this may be the dynamical stability of the inverse pendulum with the vibrating support point, which is described by the equation (1) provided that the angle X is small. Simple considerations show that at large frequency w the pendulum support point vibration becomes equivalent to the effective restoring force directed towards the vertical. When the pendulum rod is accelerated downwards, the angle X is decreased by X1. The force moment is also decreased. In the upward movement the effect is analogous, except that the angle (X-X1) is increased only by X2 < X1. In spite of the fact that the vertical acceleration is alternating, it results in the force pulling the pendulum rod back to the vertical line and thereby decreasing the angle X. Thus, the pendulum will become stable in the position opposite to the normal one. Under the resonance condition the system described by the equation (1) loses its stability and the amplitude of vibrations tends to infinity. The possibility of the stable state of the inverse pendulum was probably first pointed out by B. Van der Pol in 1925. In 1950 P.L. Kapitsa elegantly described this phenomenon using the method of approximate solution and demonstrated experimentally the stable inverse pendulum ("Kapitsa pendulum"). Furthermore, he set the problem of the observable dynamical stable states of particles and molecules. In 1958 M.L. Gaponov, M.A. Miller gave a theoretical proof of the existence of potential wells in the system of charged particles placed in the inhomogeneous high-frequency electromagnetic field and in 1959 R.F. Wuerker, H. Shelton, and R.V. Langmuir experimentally demonstrated the levitation effects in the system of charged particles moving simultaneously in both the inhomogeneous constant and inhomogeneous variable electric fields beyond the resonance conditions. In 1974 H. van der Heide made similar experiments on particles having magnetic moments and moving simultaneously in both the constant and variable magnetic fields beyond the resonance conditions. Besides, he also proposed some practical applications to the above effects (in particular, the elementary particle physics and the development of magnetically levitated trains). Contrary to the opinion published in the paper "On the problem on levitating coffins" H. Van der Heide remarked the feasibility of the steady movement of the constant magnet in the field of another constant magnet. It should be noted, that in general it is practically impossible to find the solution of the non-linear system of equations, describing the motion of the many-body system, when all of the translational and rotational degrees of freedom are taken into account. Therefore most of the researchers used to confine themselves to the analysis of simple linear systems far from the resonance conditions. Only V.V. Kozorez in 1981 considered a more complicated non-linear dynamic system of the magnetically interacting free bodies. Historically the problems, which were considered first, were the problem of the motionless resonator and the problem of the single-frequency resonator. Later on it was found that the problem of the motion of a body under the conditions of magnetic resonance is of special novelty (A.I. Filatov, V.G. Shironosov). The peculiarity of the magnetic resonance consists in the dependence of the resonance frequency on the translational and rotational degrees of freedom of the sample imposed by the inhomogeneous magnetic field and this in turn results in the appearance of the spatially resonant zones. Using analogue computer simulation, physical experiments and methods of analitical computer calculations, we could demonstrate the "impossible phenomenon" - the levitation of the sample in the resonant zone. The principle role in this phenomenon plays the non-linearity of the system. Furthermore, though it may look strange at first sight, the levitation of the constant magnet in the variable magnetic field alone both inside the resonance zone and outside of it was demonstrated experimentally . Similar experiments showed the existence of the levitation of the constant magnet imposed simultaneously by the inhomogeneous constant and inhomogeneous variable magnetic fields in the resonance zone 

---In addition, a number of effects of the resonance trapping and resonance confinement of the particles was found theoretically in both the inhomogeneous fields and in the system of two free magnetic dipoles, the spin moments of which were included in the equations of motion .