Generalized Birkhoff system and a kind of combined gradient system

A combined gradient system which is obtained by adding a gradient system to a skew-gradient system is proposed. The property of the system is studied. Three results on its solution are obtained. Moreover, a criterion on the stability of the system is presented. The generalized Birkhoff system is a more extensive constrained mechanical system than Lagrange system, Hamilton system and Birkhoff system. The conditions under which a generalized Birkhoff system can be considered as a combined gradient system are given. When a generalized Birkhoff system is transformed into a combined gradient system, its integration and stability can be discussed by using the property. Finally, some examples are given to illustrate the application of our results.