(Quasy)periodic solution in (in)finite dimensional hamiltonian systems with applications to celestial mechanics and wave equation

In this text, the authors explore the richness of Hamiltonian systems and their profound implications across various domains of physics and mathematics. The book systematically addresses the complex theory of Hamiltonian dynamics, beginning with foundational concepts and leading into advanced discussions on quasi-periodic solutions. By focusing on both finite and infinite-dimensional systems, it uniquely covers a breadth of applications from celestial mechanics to the understanding of wave equations.

Hamiltonian systems are pivotal in the field of dynamical systems and their study can reveal critical insights into the motion of celestial bodies and the behavior of waves. The book offers an in-depth mathematical treatment, making comprehensive use of advanced analytical techniques to examine the existence, stability, and structure of periodic and quasi-periodic solutions. This comprehensive approach helps elucidate how seemingly erratic motions possess underlying order and regularity.

The text delves into the application of these mathematical constructs in celestial mechanics, addressing problems such as the restricted three-body problem, and the long-term evolution of planetary systems. For wave equations, the book explores the implications of Hamiltonian structure in understanding wave propagation in various media.

• In-depth insight into the mathematical structures underpinning Hamiltonian systems, offering methods to find and analyze solutions.
• Understanding the interplay between stability and chaos in dynamical systems through the lens of periodic and quasi-periodic motions.
• Application of theoretical findings to real-world phenomena, particularly in the realms of celestial mechanics and wave propagation.
• Advanced mathematical techniques, particularly those involving analytic functions and perturbation methods, are developed and applied.

"The elegance of Hamiltonian systems lies not only in their conservation laws but in the profound symmetry that governs both motion and rest."

"Among the stars and waves, it is the mathematical truths inscribed within Hamiltonian dynamics that reveal the hidden order of nature."

"Quasi-periodicity is the silent song of the cosmos, resonating through the orbits of planets and the vibrations of waves."

This book stands at the intersection of pure mathematics and applied physics, an amalgam that seeks to further our understanding of the natural universe through the power of mathematical theory. The relevance of Hamiltonian systems in physics cannot be overstated. They find application in various disciplines, from analyzing the long-term behavior of the solar system to appreciating the fundamental nature of wave phenomena. By capturing both the theoretical and practical aspects, this book not only serves as a vital resource for researchers and students in these fields but also as a cornerstone for anyone interested in the deeper mathematical truths that govern dynamical systems.

The book is essential for developing a rigorous understanding of the principles that govern the predictable unpredictability of Hamiltonian systems, and it sets the stage for future inquiries into the enduring mysteries of dynamical behavior in the cosmos and the continuous mediums through which waves travel.