Vladimir Ol'shanskii

For Poincaré–Zhukovsky’s equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $({\bf YS},{\bf K}) = \rm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky’s case.

In this paper, the conditions of nonregular precession with a constant ratio of the velocities of precession and proper rotation for a gyrostat in the superposition of two homogeneous and one axisymmetric field are obtained. The case where the gyrostat has axial dynamical symmetry and the proper rotation axis coincides with the body’s symmetry axis is singled out. It is shown that in the case where the gyrostatic momentum is collinear to the symmetry axis, nonregular precession is possible with a precession velocity equal to, twice as large as, or twice as small as the proper rotation velocity. In each of these cases, the condition expressing the ratio of the axial and equatorial inertia moments of the body in terms of the nutation angle coincides with the corresponding condition obtained earlier for the nonregular precession of a solid in three homogeneous fields. In the particular case of the gyrostat’s spherical symmetry, when the precession speed is half or twice as large as its proper rotation speed, the cosine of the nutation angle is equal to one fourth; at equal speeds, the nutation angle should be equal to sixty degrees. The sets of admissible positions of the forces’ centers for the general case of nonorthogonal fields are found. The precession of a gyrostat whose gyrostatic momentum is deflected from the symmetry axis is considered. The possibility of nonregular precession is shown for the case where the precession velocity is twice as large as the proper rotation velocity. The solution is expressed in terms of elementary functions. The rotation of the gyrostat is either periodic or the rotation velocity tends to zero and the carrier body of the gyrostat approaches the equilibrium position.