Contents

Preface

1 Symplectic manifolds and symplectic relations
1.1 Symplectic manifolds
1.2 Symplectic vector spaces1.3 Special submanifolds
1.4 The characteristic foliation of a coisotropic submanifold

1.5 Relations

1.6 Symplectic relations
1.7 Linear symplectic relations
1.8 Symplectic reductions
1.9 Symplectic relations generated by a coisotropic submanifold
1.10 The symplectic background of the Cauchy problem

2 Symplectic relations on cotangent bundles
2.1 Cotangent bundles
2.2 The canonical symplectic structure of a cotangent bundle
2.3 One-forms as sections of cotangent bundles
2.4 Lagrangian singularities and caustics
2.5 Generating families
2.6 Generating families of symplectic relations
2.7 The composition of generating families
2.8 The canonicallift of submanifolds
2.9 The canonicallift of relations

3 The geometry of the Hamilton-Jacobi equation.
3.1 The Hamilton-Jacobi equation
3.2 Characteristics and rays
3.3 Systems of rays and wave fronts
3.4 The characteristic functions
3.5 Sources, mirrors, lenses
3.6 The theorem of Jacobi
4 Hamiltonian optics in Euclidean spaces
4.1 The distance function

 

VIII Conte

4.2 From wave optics to geometrical optics
4.3 The eikonal equation and the global Hamilton principal function
4.4 The eikonal equation in a space of constant negative curvature

5 Control of static systems
5.1 Control relations
5.2 Simple c10sed thermostatic systems
5.3 The ideaI gas

5.4 Control modes and the Legendre transformation
5.5 Thermostatic potentials
5.6 Simple open thermostatic systems
5.7 Composite thermostatic systems

6 Supplementary topics 6.1 Symplectic relations generated by a submanifold
6.2 The canonicallift of reductions and diffeomorphisms
6.3 Basic observables
6.4 Canonicallift of vector fields
6.5 Regular distributions and Frobenius' theorem
6.6 Exact Lagrangian submanifolds
6.7 Dual pairings
6.8 Lagrangian splittings and canonical basis

Appendix A. Notation and basic notions of calculus on manifolds
A.l Tangent vectors and tangent bundles
A.2 The tangent functor
A.3 Special mappings
A.4 Submanifolds
A.5 Vector fields
A.6 IntegraI curves and flows
A.7 First integrals
A.8 Lie bracket
A.9 One-forms
A.10 Exterior forms
A.11 Exterior algebra
A.12 Pull-back
A.13 Derivations
A.14 The differential
 A.15 Interior product
A.16 Lie derivative

Appendix B. Global Hamilton principal functions of the eikonal equations on S2 and H2 by S. Benenti and F. Cardin
B.1 Introduction
B.2 Vector calculus in the real three-space
B.3 The Hamilton principal function of §2 .
B.4 The Hamilton principal function of lHI2
Bibliography

Index