On stability of stationary motion of mechanical conservative system

The article studies the stability of stationary motion of a nonlinear mechanical conservative autonomous system that rotates a rigid body around a fixed point. For the system under study, the first three common integrals are known: energy, moment of momentum, Poisson. At Appelrot equality that connects the moments of body inertia with the coordinates of the center of mass, a particular Hess integral is allowed. The steady-state motion under research is also valid for the existence of the Hess integral. The research of stability is carried out by the equations of linear approximation of perturbed motion. It is based on the existence of only zero and purely imaginary roots of the characteristic equation with the corresponding simple elementary divisors. This is expressed by a system of three inequalities from the coefficients of the characteristic equation, moreover, the double zero root has simple elementary divisors. The analysis of three inequalities expressing purely imaginary simple roots of the characteristic equation, enabled us to identify seven areas of solutions. The cases of degeneration of the characteristic equation are separately considered: the appearance of additional zero and multiple purely imaginary roots. Thus, the instability in the linear approximation is revealed under the condition of the existence of a particular Hess integral. The necessity of using the system of analytical computations is shown.