The Variational Theory

In this chapter we introduce the variational theory based on the action functional ( 1.1 ) in Chap. 1 . The theory has the remarkable advantage that it is nonperturbative since it applies to all convex Lagrangian systems, not necessarily nearly integrable, and also it is global in the sense that some important dynamical invariant objects can be found for all parameters (homology and cohomology classes), in particular, those parameters corresponding to the complement of the KAM tori. Thus, the theory is an important development after the KAM theory and the hyperbolicity theory. The development is parallel to similar themes in elliptic PDE and geometric analysis, where people first look for weak solutions then try to improve the regularity of the solutions. However, it is worth of pointing out that the theory in this chapter is mainly a framework, which in general does not provide concrete information for a given concrete system.