On stability of one steady-state motion of a mechanical system with the Hess partial integral

Rotation of a rigid body about a fixed point, which is described by firstorder differential equations, is considered. A large number of the first integrals are of interest from the viewpoint of investigation of mechanical systems. As far as conservative autonomous systems characterized by three degrees of freedom are concerned, both under integration and investigation of the principal dynamic properties, it is sufficient to have four integrals independent of time, which may be both general and partial. Systems with the Hess partial integral attracted great attention from the viewpoint of such an investigation earlier, and still these systems attract attention at present.. In the present paper, investigation of stability of one type of steady-state motions of a mechanical system assuming the Hess partial integral with the aid of Lyapunov’s second method was conducted. The Lyapunov function is constructed according the Chetayev’s method by the bundle of first integrals of perturbed motion. In the process of analysis, removing of some part of the variables was preliminarily fulfilled. These variable represent deviations from the steady-state motions, from the first integrals with fixed constants. As regards the quadratic expression, removing the variables is conducted by expanding in uprising series. Investigation of positive definiteness of the non-homogeneous Lyapunov function was conducted according to the criterion of sign-definiteness of polynomials of many variables. In the aspect of analysis, necessitated was a large number of various operations bound up with symbolic information processing, which were executed by the system of analytical computations installed on the personal computer. As a result of the computations conducted, revealed was formal conventional stability for the terms up to the forth order inclusively.