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.