Normal Forms and Stability of Hamiltonian Systems: (218 Applied Mathematical Sciences)

This book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics. This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of stronglystable systems. With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.

Table of Contents:
Foreword.- Preliminaries on Advanced Calculus.- Hamiltonian Systems Theory.- Normal Forms of Hamiltonian Systems.- Spectral Decomposition of Hamiltonian Matrices.- The General Linear Normalization.- Stability of Equilibria.- Stability of Linear Hamiltonian Systems.- Parametric Resonance.- References.- Index.

About the Author :
Hildeberto Cabral is an Emeritus Professor at the Federal University of Pernambuco, Brazil. He did his PhD at the University of California, Berkeley (1972), after getting a Master's degree from the Institute of Pure and Applied Mathematics–IMPA, Brazil. He does research on dynamical systems, focusing on Hamiltonian systems, celestial mechanics, stability of equilibria, and periodic solutions. Lúcia Brandão Dias is an Associate Professor at the Federal University of Rondônia, Brazil. She holds a PhD in Mathematics (2007) from the Federal University of Pernambuco, Brazil, with post-doc studies at the same university. Her research interests lie in Hamiltonian systems, differential equations, and n-body problems.

Review :
“The book is self-contained. … This comes out in the superb presentation of the material, with plenty of examples that illustrate the concepts and the methods surveyed throughout the text. … The book ‘can be used both as a textbook for courses at the graduate level and as a reference book for researchers who need these techniques in dealing with the analysis of stability of equilibria in problems of classical mechanics’.” (Juan R. Pacha, Mathematical Reviews, December, 2024) “Hamiltonian systems, their stability and their normal forms are the main topics of this book. These are presented in an historical context that gives the reader a good sense of their development over time while identifying the major contributors to the theory. Authors’ preface does a particularly good job here – identifying the origins and motivations of the ideas and describing how all the various parts – concepts, methods and tools – fit together.” (William J. Satzer Jr., zbMATH 1533.37001, 2024)