Applications of variational methods to Hamiltonian systems

In this thesis we study three different problems in Hamiltonian systems using a variational approach. The first problem we study is the existence of multiple solutions of Lagrangean systems having the n-dimensional torus as the configuration space. The multiplicity results are obtained by using the Ljusternik-Schnirelmann relative category. The idea is to relate the level sets of the potential energy on ;The second problem is the study of existence and multiplicity of T-periodic solutions of a Hamiltonian system, where the Hamiltonian is periodic in the space variable q, and superlinear in the momentum variable p. We give results for both forced and unforced systems. The basic existence theorem used here is a generalization of the Saddle Point Theorem. It considers a functional on E x M, where E is a Hilbert space and M is a compact manifold. The Saddle Point condition is assumed on E, uniformly with respect to M.;The third problem we studied is the existence of subharmonic solutions converging to an equilibrium point for a Hamiltonian system. This problem is analyzed using global arguments by extending the Hamiltonian to the whole space. The existence of subharmonics is obtained using the generalized Mountain Pass Theorem. The hypothesis on the Hamiltonian are of two kinds. For the first case we assume a condition on the quadratic art in terms of the Floquet exponents of the corresponding linearized system. This condition relaxes hypothesis considered by other authors. The second case we consider is when the higher order term is superquadratic and we show that with this condition the problem can be reduced to the center manifold of the system.