On the stability of the equilibrium for a mechanical system with dissipation

A.V. Fedotov

Moscow Institute of Electronic Engineering

103498, Moscow, Zelenograd, MIEE

We consider a mechanical system with Lagrangian function in a neighbourhood of the critical point of the potential energy . It is well known that, when the critical point of the potential energy is a strict local minimum, the proper equilibriumis stable. The classical Kelvin theorem states that this equilibrium does not loss stability, if the dissipative forces are included in consideration. As it was shown by V.V. Kozlov, if has no minimum at the origin, and the absence of minimum is determined by the first nontrivial homogenious form in the Maclaurin expansion of V, then the equilibrium is unstable. The instability is a consequence of the existence of an asymptotic solution: as , .

In the case, when the minimum absence is established by the quadratic form , the existence of such a solution can be proved with the help of Lyapunov`s first method. Therefore, we set that . 

One part of the recent work represents the further investigation of this Lagrangian dynamic system, however, we seek the asymptotic solutions for the system with partial dissipation. In another section, we examine ability of the equilibrium stabilization of the nonholonomic system.

1. Nonholonomic system with dissipation. We consider the Lagrange equations 

(1)

in the small neighbourhood of the equilibrium .

Here is a kinetic energy of the system, is the Maclaurin expansion of a potential energy, with a homogeneous form of degree s and , is a dissipative Rayleighfunction (). Nonholonomic constraint is determined by the second equation (1). Let note the plane in the n-dimensional space .

Theorem 1. If the restriction of to the has no minimum at , then equation (1) has a solution asymptotically tending to the equilibrium , i. e. , as .

In particular, we get that the equilibrium point is unstable. The proof of the theorem is based on the consequent constructing a formal series solution

and on the Kuznetsov theorem, which guaranteed that actual solution, having this formal series as asymptotic expansion, exists.

2. The holonomic system with the partial dissipation. In this section, the existence of the asymptotic solution of the equation 

(2)

is under investigation. However, in this case, the matrix of quadratic form is degenerated. It is supposed, that is a r-dimensional plane.

Theorem 2. If the restriction to the plane has no minimum at the origin, then the equilibrium is unstable.

The theorem is proved also by constructing solution, wich asymptotically tend to the equilibrium. If the conditions of theorem 2 are fulfilled, then equation (2) has the solution , with the following asymptotic series forand 

,,

where the coefficients ,depend on as polinomials.