Analysis On The Solutions Of Variational Method And Differential Equation System

Variational methods concentrate on solving extremal problems of functionals. By the efforts of generations of mathematicians, people find the relations between solving differential equations by their generalized solutions and extremal problems of functionals. Therefore the central task of modern calculus of variations develops to construct appropriate functionals, and then analize solutions of equations by the critical points of functionals. It is often difficult to solve equations explicitly, but we could know information about the solutions by the critical points of functionals.The existence of solutions of the extemal problems of functionals relies heavily on the domain of the functionals. We first introduce some basic knowledge on spaces of functions to make use of variational methods. Then we discuss the Fréchet differentiability of three different types of functionals. For the minimax methods guarantee the existence of critical points of the saddle type, and the mountain pass lemma is just an important result to use. Then we explore sufficient conditions for Hamiltonian systems of separated variables type to satisfy the mountain pass lemma,and we conclude the existence of homoclinic solutions of a big class of differential equations.At last we explore general theory of planar integrable systems and Hamiltonian systems, and obtain some different results such as sufficient conditions to distinguish singular points of the central type. And then we use the classical Fr?mmer methods to study dynamics of planar Hamiltonian systems of separated variables type near isolated singular points.