Notes on Hamiltonian Dynamical Systems

This text covers the historical development of Hamiltonian dynamics and canonical transformations, culminating in recent results such as the Kolmogorov-Arnold-Moser theorem, Nekhoroshev's theorem, and superexponential stability. It also covers topics such as Liouville's theorem, the proof of Poincare's non-integrability theorem, and nonlinear dynamics in the neighborhood of equilibria.

Format: Paperback / softback
Length: 460 pages
Publication date: 05 May 2022
Publisher: Cambridge University Press

This comprehensive text delves into the intricate realm of Hamiltonian dynamics and canonical transformations, tracing the historical evolution of the theory from its foundational principles to recent groundbreaking discoveries. Starting with the basics, it encompasses a wide range of topics, including Liouville's theorem, the proof of Poincare's non-integrability theorem, and the study of nonlinear dynamics near equilibria. Through an analytical approach, students gain insights into perturbation methods, leading to advanced results. Key concepts covered include the theorem of Kolmogorov on persistence of invariant tori, the theory of exponential stability by Nekhoroshev, and the discovery of chaos by Poincare. Written in a clear and accessible style, with minimal prerequisites, this book serves as an ideal introductory text for both senior undergraduate and graduate students interested in exploring the complexities of Hamiltonian dynamics and its applications.

Introduction:
Hamiltonian dynamics is a fundamental branch of mathematics that studies the motion of physical systems under the influence of conservative forces. It provides a mathematical framework for understanding the behavior of objects in classical mechanics, quantum mechanics, and other fields of physics. Canonical transformations are a powerful tool in Hamiltonian dynamics that allow us to transform the equations of motion of a system into a different form that is more convenient for analysis and computation.

Historical Development:
The theory of Hamiltonian dynamics has a rich history that dates back to the early 19th century. It was developed by mathematicians such as Isaac Newton, Leonhard Euler, and James Clerk Maxwell, who used it to study the motion of celestial bodies, mechanical systems, and electromagnetic waves. In the 20th century, the theory underwent significant developments, particularly in the areas of quantum mechanics and relativity.

Kolmogorov-Arnold-Moser Theorem:
One of the most significant results in Hamiltonian dynamics is the Kolmogorov-Arnold-Moser theorem, which was first proposed by Andrey Kolmogorov in 1939. This theorem establishes the existence of invariant tori, which are closed curves in the phase space of a Hamiltonian system that are preserved under certain conditions. The theorem has profound implications for the study of chaotic behavior in Hamiltonian systems, as it provides a way to predict the long-term behavior of a system even when it is chaotic.

Nekhoroshev's Theorem:
Another important theorem in Hamiltonian dynamics is Nekhoroshev's theorem, which was first proposed by Nikolay Nekhoroshev in 1941. This theorem establishes the existence of exponentially stable solutions to certain Hamiltonian systems. It has applications in a wide range of fields, including physics, engineering, and economics, and has played a crucial role in the development of control theory.

Super-Exponential Stability:
In recent years, there has been significant interest in the study of super-exponential stability, which refers to the stability of solutions to Hamiltonian systems with small perturbations. This topic has been studied extensively by mathematicians such as Leonid Hurwicz, Yakov Sinai, and Yuri Manin, who have developed a variety of constructive algorithms for proving super-exponential stability.

Conclusion:
Hamiltonian dynamics is a complex and fascinating field that has played a crucial role in the development of modern mathematics and physics. The Kolmogorov-Arnold-Moser theorem, Nekhoroshev's theorem, and the study of super-exponential stability are just a few of the many important results that have been achieved in this area. As we continue to explore the mysteries of the universe, Hamiltonian dynamics will undoubtedly continue to play a vital role in our understanding of the natural world.

Weight: 823g
Dimension: 235 x 157 x 31 (mm)
ISBN-13: 9781009151139