On the stability of one type of permanent rotations of a mechanical system with a Hessian paticular integral

The paper considers a mechanical autonomous conservative system with a Hessian particular integral in the well-known problem of the rotation of a rigid body around a fixed point. This system is described by six ordinary differential equations. We have carried out a study of the Lyapunov stability of four types of stationary motions for which all components of the Poisson angles are nonzero. The proposed stationary movements are permanent rotations. The study is based on the first Lyapunov method. For this purpose, we derive a characteristic equation for the differential equations of disturbed motion on the matrix of the right-hand side in the linear approximation. Obtaining algebraic expressions, their simplification and factorization are carried out by a system of analytical calculations on modern personal computers. The same characteristic equations are established for all permanent rotations. The analysis of the coefficients of the obtained equations made it possible to establish, in addition to zero and purely imaginary roots, the presence of two real solutions other than zero. In conservative systems, one of them will be positive. The calculations performed showed the instability of all the studied permanent rotations.