Lyapunov-Kozlov method for singular cases

Abstract

Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.

Analysis of the motion and stability of the holonomic mechanical system in the arbitrary force field

“…Moreover, stability of the specific mechanical system was discussed using different approaches. Namely, unlike the papers [8] and [9], where the relative advantages and disadvantages of various analytical methods of nonholonomic systems are briefly presented, the problem of the instability of the equilibrium state of a scleronomic mechanical system with linear homogeneous constraints are considered in [10], and the problem of the stability of the equilibrium state in the case with holonomic mechanical systems in [11].…”

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

“…Kozlov's generalization (see [1][2][3][4]) of the Lyapunov first method [5] for the case of strongly nonlinear systems of differential equations of motion allows solving the problem of instability of equilibrium position of a mechanical system also for the cases when the conditions of existence and uniqueness of solutions of differential equations of motion (see [6,7]) are not satisfied.…”