It is a difficult problem to study the stability of the rheonomic and nonholonomic mechanical systems. Especially it is difficult to construct the Lyapunov function directly from the differential equation. But the gradient system is exactly suitable to study the stability of a dynamical system with the aid of the Lyapunov function. The stability of the solution for a simple rheonomic nonholonomic constrained system is studied in this paper. Firstly, the differential equations of motion of the system are established. Secondly, a problem in which the generalized forces are exerted on the system such that the solution is stable is proposed. Finally, the stable solutions of the rheonomic nonholonomic system can be constructed by using the gradient systems.
All types of gradient systems and their properties are discussed. Two problems connected with gradient systems and mechanical systems are studied. One is the direct problem of transforming a mechanical system into a gradient system, and the other is the inverse problem, which is transforming a gradient system into a mechanical system.
The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.
A gradient system and a skew-gradient system can be merged into a combined gradient system. The differential equations of the combined gradient system are established and its property is studied. If a mechanical system can be represented as a combined gradient system, the stability of the mechanical system can be studied by using the property of the combined gradient system. Some examples are given to illustrate the applications of the results.
The skew-gradient representation of a generalized Birkhoffian system is studied. A condition under which the generalized Birkhoffian system can be considered as a skew-gradient system is obtained. The properties of the skew-gradient system are used to study the properties, especially the stability, of the generalized Birkhoffian system. Some examples are given to illustrate the application of the result.
