On a Case of Periodic Motions of the Lagrangian Top with Vibrating Fixed Point $S^2$

The motion of the Lagrangian top, the fixed point of which performs small-amplitude vertical oscillations, is studied. The case, when values of constants $a$ and $b$ of cyclic integrals are related by $|a|=|b|$, is considered. It is known that for the classical Lagrangian top with a fixed point there exists either one regular precession of the top or none at $a=b$ depending on values of parameters of the problem (the parameter a and the parameter, characterizing the position of the center of mass of the top on the axis of symmetry); at $a=-b$ there is no regular precessions. The existence of periodic motions of the top close to regular precessions (with a period equal to the period of oscillations of the fixed point) is investigated in case of oscillations of the fixed point. In the first part of the present paper periodic motions of the top, which occur from regular precession, are found in case $a=b$ and stability of these motions is determined. In the second part it is additionally assumed that oscillations of the fixed point have a high frequency, and angular velocities of precession and self-rotation of the top are small. A qualitatively new result is obtained which does not have an analogue either in the classical problem or in a problem of motion of the top with small-amplitude oscillations of the fixed point: Motions of the top close to regular precessions are found, these motions exist both at $a=b$ and at $a=-b$, and their number is different (there are two, one or none at $a=b$, and there are two or none at $a=-b$) depending on values of parameters of the problem. The problem of stability of these motions is solved precisely with the help of the KAM-theory.