Critical Point In The Second Order Hamiltonian Systems

Variational principle is a common principle in the world of the nature, and many problems can be reduced to the problems of critical points of a special fuctional under some specific conditions. Variational methods of differential equations changes the boundary value problems of differential equations into variational problems to gain the existence and numbers of the solutions to differential equations. Modern variational methods has developed greatly in recent twenty years, which is widely used in all branches of mathematics.The whole paper contains four chapters, and the periodic solutions of some nonautonomous second order Hamilton systems with suitable conditions are studied through minmax theorems in variational methods.Chapter 1 is devoted to the evolution and development of the principle of the calculus of variations and the problems that will be studied are presented.In chapter 2, some essential definitions and preliminary theorems concerning Hamilton variational methods are introduced.Chapter 3 is devoted to the application of the minmax principle on second order systems introducing the saddle point theory and some other minmax methods and some recent results are obtained by minmax methods. In this chapter, the saddle point theory is mainly used to the studies of some class of second order Hamilton systems.Chapter4 summaries some theorems gained in this article systematically and presents some problems which need further studies.