On invariant sets for the equations of motion of a rigid body in the Hess–Appelrot case

We consider the problem of motion of a rigid body in the Hess–Appelrot case when the equations of motion have three first integrals as well as the invariant manifold of Hess. On the basis of the Routh–Lyapunov method and its generalizations, the qualitative analysis of the above equations written on the manifold is done. Stationary invariant sets for the equations are found and their Lyapunov stability is investigated. By stationary sets, we mean sets which consist of the trajectories of the equations of motion and possess the extremal property: the necessary extremum conditions for the elements of the algebra of problem's first integrals are satisfied on them. In this paper, an extension of the technique for finding such sets is proposed: obtaining new sets from previously known ones and by means of “the inverse Lagrange method”. Applying these techniques, we have found a family of invariant manifolds for the differential equations on the invariant manifold of Hess. From this family, several invariant manifolds of greater dimension than those of the family have been obtained, and an analysis of differential equations on one of them was done. Equilibrium positions and families of permanent rotations of the body have been found. For a number of the solutions, sufficient stability conditions have been derived, including with respect to part of variables.