On the instability of equilibrium position of a mechanical system with singular constraints

Abstract

Key words Lyapunov first method, nonholonomic system, singular constraints, instability of equilibrium.The Lyapunov first method generalized to the case of nonlinear differential equations is applied to the study of the instability of the equilibrium position of a mechanical system, whose motion is constrained by singular nonholonomic constraints. Starting from the results of S. D. Furta (On the instability of equilibrium position of constrained mechanical systems) three theorems on the instability are formulated. The first theorem considers the case of nonholonomic constraints that do not satisfy the condition of weak nonholonomity. The other two theorems are related to the case of weakly nonholonomic systems. In each of the formulated theorems it is shown that the minimum form of Maclaurin series for the potential energy has not a local minimum. Thus, a contribution has been made to the inversion of Lagrange's theorem.