Normal Forms and Stability of Hamiltonian Systems

Hildeberto E. Cabral, Lúcia Brandão Dias

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English

Springer International Publishing AG
12 August 2023

Cybernetics & systems theory; Calculus & mathematical analysis; Differential calculus & equations; Non-linear science; Chaos theory; Mathematical physics; Dynamics & vibration

Series: 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.

By: Hildeberto E. Cabral, Lúcia Brandão Dias
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2023 ed.
Volume: 218
Dimensions: Height: 235mm, Width: 155mm,
Weight: 705g
ISBN: 9783031330452
ISBN 10: 3031330455
Series: Applied Mathematical Sciences
Pages: 337
Publication Date: 12 August 2023
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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.

Reviews for Normal Forms and Stability of Hamiltonian Systems

“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)