Two kinds of generalized gradient representations for generalized Birkhoff system

Brikhoff system is a kind of basic dynamical system. The theory and method of Brikhoff system dynamics have been applied to the hadron physics, quantum physics, relativity and rotational relativistic system. The properties of gradient system not only play an important role in revealing the internal structure of dynamical system, but also help to explore the dynamical behavior of the system. In this paper, two kinds of generalized gradient representations for generalized Birkhoff system are studied. First, two kinds of generalized gradient systems, i.e., the generalized skew gradient system and the generalized gradient system with symmetric negative definite matrix, are proposed and the characteristics of the systems are studied. Second, the relations of stability between these two kinds of gradient system and the dynamical system are discussed. Third, the condition under which a generalized Birkhoff system can be considered as one of the two generalized gradient systems is obtained. Fourth, the gradient discrimination method of stability of the generalized Brikhoff system is given, and the characteristics of the generalized gradient systems can be used to study the stability of the generalized Birkhoff system. Finally, some examples are given to illustrate the application of the result. Therefore, once the mechanical system is expressed as the generalized gradient system, the stability and the asymptotic stability can be conveniently studied by using the properties of generalized gradient system. The difficulty in constructing Lyapunov functions is avoided, and a convenient method of analyzing the stability of mechanical system is provided.