Lyapunov first method for nonholonomic systems with circulatory forces

We consider the problem of instability of equilibrium states of scleronomic nonholonomic systems moving in a stationary field of conservative and circulatory forces. The applied methodology is based on the existence of solutions of differential equations of motion which asymptotically tend to the equilibrium state of the system. It is assumed that the forces in the neighbourhood of the equilibrium position can be presented in the form of the sum of two components, the first one being a homogeneous function of the position with the positive degree of homogeneity; the second one being infinitely small in comparison to the first one. The results obtained, which partially generalize results from [S.D. Taliaferro, Instability of an equilibrium in a potential field, Arch. Ration. Mech. Anal. 109 (2) (1990) 183–194; V.A. Vujičić, V.V. Kozlov, Lyapunov’s stability with respect to given state functions, J. Appl. Math. Mech. 55 (4) 9 (1991) 442–445; D.R. Merkin, Introduction to the Theory of the Stability of Motion, Nauka, Moscow, 1987 (in Russian); A.V. Karapetyan, On stability of equilibrium of nonholonomic systems, Prikl. Mat. Mekh. 39 (6) (1975) 1135–1140 (in Russian)], are illustrated by an example.