The N-Vortex Problem

Preface.- 1 Introduction.- 1.1 Vorticity Dynamics.- 1.2 Hamiltonian Dynamics.- 1.3 Summary of Basic Questions.- 1.4 Exercises.- 2 N Vortices in the Plane.- 2.1 General Formulation.- 2.2 N = 3.- 2.3 N = 4.- 2.4 Bibliographic Notes.- 2.5 Exercises.- 3 Domains with Boundaries.- 3.1 Green's Function of the First Kind.- 3.2 Method of Images.- 3.3 Conformai Mapping Techniques.- 3.4 Breaking Integrability.- 3.5 Bibliographic Notes.- 3.6 Exercises.- 4 Vortex Motion on a Sphere.- 4.1 General Formulation.- 4.2 Dynamics of Three Vortices.- 4.3 Phase Plane Dynamics.- 4.4 3-Vortex Collapse.- 4.5 Stereographic Projection.- 4.6 Integrable Streamline Topologies.- 4.7 Boundaries.- 4.8 Bibliographic Notes.- 4.9 Exercises.- 5 Geometric Phases.- 5.1 Geometric Phases in Various Contexts.- 5.2 Phase Calculations For Slowly Varying Systems.- 5.3 Definition of the Adiabatic Hannay Angle.- 5.4 3-Vortex Problem.- 5.5 Applications.- 5.6 Exercises.- 6 Statistical Point Vortex Theories.- 6.1 Basics of Statistical Physics.- 6.2 Statistical Equilibrium Theories.- 6.3 Maximum Entropy Theories.- 6.4 Nonequilibrium Theories.- 6.5 Exercises.- 7 Vortex Patch Models.- 7.1 Introduction to Vortex Patches.- 7.2 The Kida-Neu Vortex.- 7.3 Time-Dependent Strain.- 7.4 Melander-Zabusky-Styczek Model.- 7.5 Geometric Phase for Corotating Patches.- 7.6 Viscous Shear Layer Model.- 7.7 Bibliographic Notes.- 7.8 Exercises.- 8 Vortex Filament Models.- 8.1 Introduction to Vortex Filaments and the LIE.- 8.2 DaRios-Betchov Intrinsic Equations.- 8.3 Hasimoto's Transformation.- 8.4 LIA Invariants.- 8.5 Vortex-Stretching Models.- 8.6 Nearly Parallel Filaments.- 8.7 The Vorton Model.- 8.8 Exercises.- References.

For applied mathematicians, physicists, and engineers interested in either nonlinear dynamics or classical mechanics and fluid dynamics. Describes the Hamiltonian aspects of vortex dynamics so that it may serve as an entry into the large literature on integrable and non-integrable vortex problems.

From the reviews:

ZENTRALBLATT MATH

"Exercises given at the end of each chapter could be very useful to the readers to enhance their knowledge...the first three chapters deal with the basic classical two-dimensional point vortex theory, and the rest of the book takes care of the recent applications and extensions of the basic theory. The references are adequate for further study. Many open problems associated with N-vortex motion are also listed. The book is a welcome addition to the book shelves of researchers pursuing the N-vortex problem."

MATHEMATICAL REVIEWS

"Although several other books on vortex dynamics have appeared in recent years, none have the level of detail on these discrete vortex models that Newton achieves. The book is sure to be a key reference work on this area for many years to come. Students of fluid mechanics will find much valuable material here on modern dynamics applied to problems of interest to them. Applied mathematicians will find an entree to frontline problems of fluid mechanics using tools with which they are eminently familiar."

"This is a very timely addition to the literature. ... Exercises are scattered throughout the book, many of them quite substantial. ... A massive bibliography - a treasure in itself - concludes the text. Although several other books on vortex dynamics have appeared in recent years, none have the level of detail on these discrete vortex models that Newton achieves. The book is sure to be a key reference work on this area for many years to come." (Hassan Aref, Mathematical Reviews, Issue 2002 f)

"The goal of this book is to describe the Hamilton aspects of vortex dynamics in such a way that graduate students and researchers can use this book as an entry level text to the rather large literature on integrable and non-integrable vortex problems ... . The book is well written with each chapter containing useful biographical notes and exercises. Of