Stability of the uniform rotation of a gyrostat round the vertical main axis on an absolutely smooth horizontal plane
The motion of a gyrostat on an absolutely smooth plane is discussed. A Hamilton function which gives the canonical equations of motion is obtained. This admits of particular solutions, namely uniform rotations round a vertical axis which are identical with that of the uniform rotations of the rotor. A transition to a system with two degrees of freedom is realized, and the expansion of the Hamiltonian in the vicinity of the corresponding position of equilibrium, with an accuracy to within fourth-order terms, is obtained. In the region of admissible values of the parameters the domain of the necessary stability conditions, and the domains where the Hamiltonian functions are of fixed sign and alternating, are examined. In those cases where the Hamiltonian is not of fixed sign, its normalization is performed, both a non-resonance situation and resonances of the first, second and fourth order being considered. The sufficient conditions for stability of uniform gyrostat rotation in terms of constraints on the coefficients of normal forms are obtained. For a clear interpretation of the results, special cases where the values of all the parameters except two are fixed, are given. The plane domain of the necessary stability conditions and resonance curves are constructed, and using computer results stability on the curves is discussed.
scientific article; zbMATH DE number 4003567