Hamiltonian Structures and Generating Families

1. Front Matter

   Pages i-xiv

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2. Basic Notions of Calculus on Manifolds

   • Sergio Benenti

   Pages 1-25

3. Relations

   • Sergio Benenti

   Pages 27-30

4. Symplectic Relations on Symplectic Manifolds

   • Sergio Benenti

   Pages 31-56

5. Symplectic Relations on Cotangent Bundles

   • Sergio Benenti

   Pages 57-76

6. Canonical Lift on Cotangent Bundles

   • Sergio Benenti

   Pages 77-88

7. The Geometry of the Hamilton–Jacobi Equation

   • Sergio Benenti

   Pages 89-119

8. Hamiltonian Optics in Euclidean Spaces

   • Sergio Benenti

   Pages 121-152

9. Control of Static Systems

   • Sergio Benenti

   Pages 153-205

10. Supplementary Topics

    • Sergio Benenti

    Pages 207-225

11. Global Hamilton Principal Functions on \(\mathbb S_2\) and ℍ₂

    • Sergio Benenti

    Pages 227-246

12. Back Matter

    Pages 247-258

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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)

• Control of static mechanical systems
• Control of thermostatic systems
• Geometrical optics
• Hamilton-Jacobi equation
• Symplectic Relations
• Symplectic geometry

From the reviews:

“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. … This new work gives an improved presentation of the theoretical part and deeper developments of its applications. … This book is aimed at both undergraduate and graduate students … in Algebra and Geometry. … Specialists will also find a nice reference in this book … .” (Marco Castrillon Lopez, The European Mathematical Society, May, 2013)
 
“The book under review is an enhanced version of a first edition in Russian; it deals with Hamiltonian structures that are often encountered in mathematical models of different physical phenomena. … The reader can find substantial bibliography and a detailed index at the end. The book is well organized, the exposition is rigorous and clear. It should attract interest of graduate students and researchers interested in the modern symplectic geometry and its applications.” (Yuri V. Rogovchenko, Zentralblatt MATH, Vol. 1233, 2012)

"…In this book the main accent is shifted to the application of modern mathematical methods of symplectic geometry and topology 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, as the presentation follows the best classical traditions of Italian geometric school (Levi-Civita, etc) and differs from the style of Russian mathematics. The … book is a significant input in the modernsymplectic geometry and its applications…” (Andrey Tsiganov, St. Petersburg State University)

• , Dipartimento di Matematica, Università di Torino, Torino, Italy

  Sergio Benenti

Sergio Benenti is a professor of mathematical physics at Università di Torino, Italy. His current fields of research include symplectic geometry with applications to physical theories, Riemannian geometry with applications to the theory of the separation of variables in the Hamilton-Jacobi equation and in other relevant differential equations of physics, and mathematical models of the dynamics of non-holonomic systems.

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