### New Treatment of the Perturbed Motions of a Rotating Symmetric Gyrostat about a Fixed Point

#### Abstract

In the present paper, we investigate the perturbed rotational motions of a symmetric gyrostat about a fixed point O, which are close to Lagrange's case. This gyrostat is acted upon by a central Newtonian force eld arising from an attracting centre $O_1$ which is located on a downward xed axis (Z- axis); a third component of the gyrostatic moment vector $\underline{l}$ about the moving axis (z- axis); restoring moment and perturbing moment vector $\underline{M}$ . The moment k is introduced to express the rotation of the body under the action of uniform magnetic field of strength $\underline{H}$ and a point charge e located on the axis of symmetry. It is assumed that the angular velocity is large, its direction is close to the axis of dynamic symmetry of the body and that two projections of the perturbing moment vector onto the principal axes of inertia of the body are small as compared to the restoring moment k while the third one is of the same order as it. A small parameter is introduced in a special way and the averaging method is used to obtain the first order approximate solutions of the equations of motion. A theoretical description for this approach in the resonant and non-resonant cases is given. The graphical representations for these solutions are presented to describe the gyrostatic motion at any time.

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