The variational functional

Abstract

In this chapter, we study finite dimensional approximation methods for Hamiltonian systems. In Section 1, we study the Galerkin approximation method for Hamiltonian systems. Then in Section 2 we define the functional corresponding to the general nonlinear Hamiltonian systems on the space L of square integrable periodic functions and study its basic properties. In Section 3 we introduce the saddle point reduction method. In Section 4 we study the dimension of the kernal of the reduced functional corresponding to the given Hamiltonian system. In Section 5, we derive certain useful estimates on the reduced functional. Results in Sections 2 to 5 form the most basic part of the saddle point reduction method.