Introduction to Hamiltonian Fluid Dynamics and Stability Theory (CRC Monographs and Surveys in Pure and Applied Math (Hardcover))

Hamiltonian fluid dynamics and stability theory work hand-in-hand in a variety of engineering, physics, and physical science fields. Until now, however, no single reference addressed and provided background in both of these closely linked subjects. Introduction to Hamiltonian Fluid Dynamics and Stability Theory does just that-offers a comprehensive introduction to Hamiltonian fluid dynamics and describes aspects of hydrodynamic stability theory within the context of the Hamiltonian formalism. The author uses the example of the nonlinear pendulum-giving a thorough linear and nonlinear stability analysis of its equilibrium solutions-to introduce many of the ideas associated with the mathematical argument required in infinite dimensional Hamiltonian theory needed for fluid mechanics. He examines Andrews' Theorem, derives and develops the Charney-Hasegawa-Mima (CMH) equation, presents an account of the Hamiltonian structure of the Korteweg-de Vries (KdV) equation, and discusses the stability theory associated with the KdV soliton. The book's tutorial approach and plentiful exercises combine with its thorough presentations of both subjects to make Introduction to Hamiltonian Fluid Dynamics and Stability Theory an ideal reference, self-study text, and upper level course book. "...provides an introduction on the field, the first book to combine both topics, it takes a tutorial approach & introduces many ideas with a simple, physical example - the non-linear pendulum." Cover Half Title Title Page Copyright Page Dedication Table of Contents About the author Acknowledgments 1: Introduction 2: The nonlinear pendulum 2.1 Model formulation 2.2 Canonical Hamiltonian formulation 2.3 Least Action Principle 2.4 Symplectic Hamiltonian formulation 2.5 Mathematical properties of the J matrix 2.6 Poisson bracket formulation 2.7 Steady solutions of a canonical Hamiltonian system 2.8 Linear stability of a steady solution 2.9 Nonlinear stability of a steady solution 3: The two dimensional Euler equations 3.1 Vorticity equation formulation 3.2 Hamiltonian structure for partial differential equations3.3 Hamiltonian structure of the Euler equations 3.4 Reduction of the canonical Poisson bracket 3.5 Casimir functionals 3.6 Noether's Theorem 3.7 Exercises 4: Stability of steady Euler flows 4.1 Steady solutions of the vorticity equation 4.2 Linear stability problem 4.3 Normal mode equations for parallel shear flows 4.4 Linear stability theorems 4.5 Nonlinear stability theorems 4.6 Andrews's Theorem 4.7 Flows with special symmetries 4.8 Exercises 5: The Charney-Hasegawa-Mima equation 5.1 A derivation of the CHM equation5.2 Hamiltonian structure 5.3 Steady solutions 5.4 Stability of steady solutions 5.5 Steadily-travelling solutions 5.6 Exercises 6: The KdV equation 6.1 A derivation of the KdV equation 6.2 Hamiltonian structure 6.3 Periodic and soliton solutions 6.4 Variational principles 6.5 Linear stability 6.6 Nonlinear stability 6.7 Exercises References Index "Hamiltonian fluid dynamics and stability theory work side-by-side in a variety of engineering, physics, and physical science fields. Until now, however, no single reference has addressed and provided background in both of these closely linked subjects. Introduction to Hamiltonian Fluid Dynamics and Stability Theory does just that - it offers a comprehensive introduction to Hamiltonian fluid dynamics and describes aspects of hydrodynamic stability theory within the context of the Hamiltonian formalism." "The book's tutorial approach and plentiful exercises combine with its thorough presentations of both subjects to make Introduction to Hamiltonian Fluid Dynamics and Stability Theory an ideal reference, self-study text, and upper-level course book."--Jacket