TITLE: The Primitive Function of an Exact Symplectomorphism AUTHOR: Alex Haro Departament de Matematica Aplicada i Analisi Universitat de Barcelona Gran Via 585 08007 Barcelona (Spain) e-mail: haro@cerber.mat.ub.es ABSTRACT: This thesis has been directed by Prof. Carles Simo. The main contribution is the systematic use of the primitive function of an exact symplectomorphism. The analytical, geometrical and numerical tools used along this thesis take into account the properties of this primitive function. In fact, they come from the structure of the phase space, given by an action form. We have divided the thesis in four parts. PART I. Exact symplectic geometry (introduction of the problems). This part contains the basic tools of symplectic geometry and outline the four subjects that we have study along the thesis: - the determination problem: although the primitive function of an exact symplectomorphism is also know as genereting function, it does not generate our symplectomorphism! - the interpolation problem: that is, to get a time-dependent Hamiltonian whose flow interpolates our symplectomorphism. - the variational problem: extend the variational principles to a broad class of symplectomorphisms (not only the monotone ones) and to a broad class of phase spaces (we avoid the use of generating functions, which do not always exist) - the breakdown problem: the study of the breakdown of invariant tori, also known as Converse KAM theory. PART II. On the standard symplectic manifold (analytical part). We recall the necessary tools to work on the standard symplectic manifold. That is, we perform a coordinate treatment of the results. First of all, we relate different kinds of generating functions to the primitive function and later we solve formally the determination problem. Then we introduce different variational principles: for fixed points, periodic orbits and orbital segments. Their invariance under certain kind of transformations of phase space is proved, and we interpret physically such results. Finally, we give the basic properties of invariant exact Lagrangian graphs, obtaining, at last, that if our graph is minimizing then its orbits are minimizing. PART III. On the cotangent bundle (geometrical part) The first three chapters are similar to the three previous ones, with the difference that we do an intrinsic treatment of the results, by considering any cotangent bundle. The fourth chapter in this part deal with the solution of the interpolation problem, given in analytic set up. PART IV. Applications (numerical part) The last part deal with the applications to Converse KAM Theory. First of all, we give a small list of different examples that we shall study later. Then, we generalize converse KAM theory by MacKay, Meiss and Stark and we related it to the Lipschitz theory by Birkhoff, Herman and Mather. Then, we perform our variational Greene method and apply it to different examples. Also we study numerically the Aubry-Mather sets in higher dimensions. After this, we apply our methods to the rotational standard map, that is a symplectic skew product. Then, we give some ideas about the geometrical obstructions for existence of invariant tori, showing them with a simple example. We also find some known Birkhoff normal forms using our methods. Finally, we explain briefly how our theory can be used for arbitrary Lagrangian foliations.