Introduction

This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. The book begins with a derivation of the vorticity equation formulation of the two dimensional Euler equations and boundary conditions appropriate for simply and multiply connected domains. It develops a general theory for infinite dimensional Hamiltonian systems of partial differential equations from an axiomatic viewpoint as motivated by the results obtained for the nonlinear pendulum. The book aims to applying the Hamiltonian formulation for the two dimensional incompressible Euler equations to the question of the stability of steady solutions. Much of classical hydrodynamic stability theory centres on looking for exponentially growing/decaying solutions to the Rayleigh stability equation. One unsatisfactory feature of the two dimensional vorticity equation is that it is, of course, incapable of modelling any three dimensional effects such as vortex tube stretching/compression.