The Routh Theorem for Mechanical Systems with Unknown First Integrals

In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane. Application of the Routh–Salvadori theorem and its generalizations [1–4] for investigation of stability of stationary motions of mechanical systems with first integrals U0 = c0, U1 = c1, . . . , Uk = ck is reduced to study the type of stationary value of U0 (here U0 can be also a nonincreasing along system trajectories function) for fixed values of U1, . . . , Uk. The effective method of such investigation is proposed in [5]. This method does not take into account equations of motion of the considered system however it is supposed that all first integrals are known explicitly. On the other hand using results by I. M. Mindlin and G. K. Pozharitskii [6] it is possible to distinguish the systems [7] for which the stability analysis does not require the explicit form of all first integrals U1 = c1, . . . , Uk = ck, except U0 = c0. Let equations of motion of a mechanical system have the following form (here T means transposition): (1) d dt (︁∂K ∂?̇? )︁ = ∂K ∂q +G?̇? − ∂W ∂q − Γ ∂W ∂p ,