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## Über dieses Buch

This book is an enhanced version of an earlier Russian edition. Besides thorough revisions, more emphasis was put on reordering the topics according to a category-theoretical view. This allows the mathematical results to be stated, proved, and understood in a much easier and elegant way.

From the reviews of the Russian edition:

"The main accent is shifted to the application . . . in geometrical optics, thermostatics and control theory, and not to the Hamiltonian mechanics only. . . . To make the book fairly self-contained, full details of basic definitions and all proofs are included. In this way, the majority of the text can be read without the prerequisite of a course in geometry. The excellent collection of examples illustrates the relatively hard and highly abstract mathematical theory and its hidden difficulties. . . . The book can rise real interest for specialists . . . . The . . . book is a significant input in the modern symplectic geometry and its applications."
(Andrey Tsiganov, St. Petersburg State University)

## Inhaltsverzeichnis

### Chapter 1. Basic Notions of Calculus on Manifolds

Abstract
It is assumed that the reader is acquainted with the notion of a real, finite-dimensional differentiable manifold (smooth manifold). The aim of this chapter is to focus on the basic tools of calculus on manifolds and on the terminology and notation adopted in this book. A particular attention is paid to the concepts of rank of a map, clean and transverse intersection of submanifolds, and derivation of exterior forms.
Sergio Benenti

### Chapter 2. Relations

Abstract
We examine the notion of “relation”, which is central to this book, at various progressive levels: relations on sets (Sect. 2.1), linear relations on vector spaces (Sect. 2.2), smooth relations and reductions (Sect. 2.3), linear symplectic relations (Sect. 3.1), symplectic relations on symplectic manifolds (Chap. 3), and symplectic relations on cotangent bundles (Chap. 4).
Sergio Benenti

### Chapter 3. Symplectic Relations on Symplectic Manifolds

Abstract
In this chapter we examine the notion of “relation” in the presence of a symplectic structure. To continue with the study of the relations between symplectic manifolds, we begin with the simplest but fundamental case of linear relations between symplectic vector spaces.
Sergio Benenti

### Chapter 4. Symplectic Relations on Cotangent Bundles

Abstract
Any cotangent bundle is endowed with a canonical structure of exact symplectic manifold. Then it becomes “natural” to apply what we have designed for the symplectic manifolds in general in the previous chapter to the special case of cotangent bundles. The material associated with this reduction becomes very rich. Among the various terms that arise spontaneously, the main one is the “generating function” of a Lagrangian submanifold, which is extended to the more general notion of “generating family”. This notion is in fact the fulcrum around which the entire analysis of the following chapters is built up.
Sergio Benenti

### Chapter 5. Canonical Lift on Cotangent Bundles

Abstract
There exists an operation, that we call canonical lift and denote by a “hat” ^, which creates “symplectic objects” on a cotangent bundle T*Q starting from “objects” on the manifold Q (vector fields,maps,submanifolds, etc.). It plays an important role in the theory of symplectic relations and in its applications. The basic lift, from which all other canonical lifts can be derived, is that of a submanifold.
Sergio Benenti

### Chapter 6. The Geometry of the Hamilton–Jacobi Equation

Abstract
A coisotropic submanifold of a cotangent bundle gives rise to several geometric objects that allow an appropriate and quite general discussion of the Hamilton–Jacobi equations. For example, the concept of “solution” appears to have two meanings: from a geometrical viewpoint, it is a Lagrangian submanifold of C (or, possibly, a Lagrangian set contained in C ), and, from an analytical viewpoint, it is a generating family satisfying a certain system of first–order PDE. One of the main problems related to a Hamilton–Jacobi equation is how to generate a (possibly unique) maximal solution from suitable initial conditions (Cauchy problem). We illustrate a geometrical construction of such a solution, by using the composition rule of symplectic relations, then we can transform this geometrical construction into an analytical method. Furthermore, other classical notions of geometrical optics, such as the system of rays and caustic of a system of rays, are more easily intelligible and manageable in a geometrical context.
Sergio Benenti

### Chapter 7. Hamiltonian Optics in Euclidean Spaces

Abstract
According to Hamilton (Hamilton 1828), a “system of rays” is a congruence of straight lines in the Euclidean three-space, orthogonal to a family of surfaces. This orthogonal integrability of the rays fails in the presence of a caustic. Moreover, a system of rays can be modified through the use of optical devices, as mirrors and lenses, or by passing through surfaces that delimit two media with different refraction index. In our approach, a system of rays without caustic is represented by a regular Lagrangian submanifold of the cotangent bundle of the Euclidean space, whereas all the optical devices are represented by symplectic relations. This chapter discusses some of the most important elementary examples.
Sergio Benenti

### Chapter 8. Control of Static Systems

Abstract
What we have done until now was created as part of geometrical optics and analytical mechanics. But, surprisingly, it can be applied to other topics of mathematical physics; for instance, the study of the behavior of static systems, purely mechanical as well as thermodynamical.
Sergio Benenti

### Chapter 9. Supplementary Topics

Abstract
This is a chapter of appendices, which develops some topics previously mentioned.
Sergio Benenti

### Chapter 10. Global Hamilton Principal Functions on $$\mathbb S_2$$ and ℍ2

Abstract
As a final argument of this book I propose a theme, simple in its formulation, but not so simple in its design: to see which are the principal Hamilton functions for the geodesics of two basic Riemannian manifolds of constant curvature. In drafting this chapter, I was pleasantly helped by Franco Cardin, University of Padua.
Sergio Benenti

### Backmatter

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