This book gives the basic definitions and results as well as some physical instances where the theory is applied, of Hamiltonian systems in terms of Lagrangian submanifolds and their generating families.
Among the brilliant developments achieved by Differential Geometry along the XX century, the geometric formulation of Classical Mechanics has proved to be one of the most relevant. In this context, the Hamiltonian formulation has attired much attention of both mathematicians and physicists. For instance, from a mathematical point of view, symplectic Geometry finds essential inspiration in Hamiltonian Mechanics whereas physicists formalize many plausible models by means of the symplectic tools studied in Geometry.
Given a manifold equipped with a symplectic structure, the submanifolds that are isotropic and coisotropic with respect to the symplectic structure are called Lagrangian. Many important concepts and definitions (or, according to many specialists, all the concepts and definitions!) in symplectic Geometry or in Hamiltonian Mechanics can be written in terms of these special submanifolds. Furthermore, if the manifold is a cotangent bundle with the canonical symplectic structure, a Lagrangian submanifold is locally generated by a function. This construction can be generalized by the so-called generating families to describe more general Lagrangian sets that are connected with some physical meaningful phenomena. The main topic of this book covers the definition, description and applications of generating families. For this purpose, the introduction of the notion of symplectic relation plays an important role.
This reference is an enhanced version of a previous book edited in Russian. This new work gives an improved presentation of the theoretical part and deeper developments of its applications. In fact, the applications of the Lagrangian submanifolds and symplectic relations given in chapters 6, 7 and 8 are especially motivating. They present the Hamilton-Jacobi theory in geometric Optics, Hamiltonian Optics in Euclidean space and control of thermostatic systems respectively. There are many other scenarios where Hamiltonian systems play a key role. A comprehensive study of them would simply overwhelm the length of a single book. From this point of view, the choice done by this book could have been different, though the importance and elegance of applications in these chapters need not further justification.
This book is aimed at both undergraduate and graduate students with just some initial knowledge in Algebra and Geometry. For this reason, the chapters try to include enough preliminaries to provide a gentle introduction to the topics covered by them. Specialists will also find a nice reference in this book specially, I think, with respect to the classical applications to Optics and thermostatics.