UK Nonlinear News, August 2000


Introduction to Hamiltonian Fluid Dynamics and Stability Theory

By G.E. Swaters

Reviewed by A.P. Fordy

Chapman and Hall, 2000,
Monographs in Pure and Applied Mathematics 102,
Hardback: ISBN 1-58488-023-6, 54.95

On the back cover of this book it is claimed that the book ``offers a comprehensive introduction to Hamiltonian fluid dynamics and describes aspects of hydrodynamic stability theory''. The motivation for the book is that until now, ``no single reference has addressed and provided background in both these closely linked subjects''.

There is certainly a need for an elementary book covering these subjects, but this is not the one! Contrary to its claim, it does not give ``a comprehensive introduction to Hamiltonian fluid dynamics'', since it just presents the two dimensional Euler and the Charney-Hasagawa-Mima equations (together with a bit of KdV theory), and these are presented in a very ad hoc way. During the last 20 years or so there have been many research papers on these topics (papers by Holm, Kuznetsov, Marsden, Zakharov and many others), whose material should be incorporated in any ``comprehensive introduction''. The student would learn a lot more by reading the book, Marsden and Ratiu, Introduction to Mechanics and Symmetry, and the recent review article, P.J. Morrison, Hamiltonian description of the ideal fluid (Rev. Mod. Phys. vol 70, 467-- 521 (1998)).

However, even if the material were deemed adequate, the presentation is not. The first technical chapter is on the nonlinear pendulum. The intention is to introduce the reader to Hamiltonian dynamics through a simple finite dimensional example. If I wished to be kind to the author I might speculate that the copy editor dropped the manuscript and collected up the papers in some random order, hurriedly sending them to the printers before being discovered! The first few items which appear in this chapter are the basic pendulum equation, followed by the Hamiltonian function, and then by the potential and kinetic energies. Next we have Hamilton's equations for the pendulum and then for a general Hamiltonian function, all without any proper definitions or discussion of phase space. We then flip to the least action principle, Lagrange's equations and the Legendre transformation, deriving Hamilton's equations long after they entered the stage! There is no logical development of the subject, which appears both confused and confusing. It is not clear whether the author assumes that the student has already attended a course on Hamiltonian dynamics or not. However, it is clear that the student would not learn the subject by reading this chapter.

This `style' is not confined to chapter 2. The entire book is disassembled in this manner. Chapters 3 and 4 ramble their way through the structure and properties of the two dimensional Euler equations. There is a bizarre derivation of the ``algebraic properties of the Jacobian'', without any reference to the canonical Poisson bracket. Poisson brackets for infinite dimensional systems are introduced as if this concept is totally independent of the previously introduced finite dimensional one. Such ideas as Poisson commutativity are reintroduced as if new! The discussion of linear versus nonlinear stability is similarly confused. It is certainly more informative to read Drazin and Reid (Hydrodynamic Stability) or the aforementioned Marsden and Ratiu.

Chapter 6 introduces us to the KdV equation. The first 3 constants of the motion are presented, but there is no general discussion of local conservation laws and constants of motion, or of commuting flows. The properties of the first Poisson bracket are laboriously checked and the second Poisson bracket briefly mentioned. Unhampered by the constraints of truth the author then states, ``The dual Hamiltonian structure of the KdV equation is characteristic of all known integrable infinite dimensional dynamical systems. It is conjectured, but at this time still unproven, that a dual Hamiltonian structure is a necessary condition for an infinite dimensional dynamical system to be integrable.'' This statement is blatantly false. The KdV equation is said to be bi-Hamiltonian because there are 2 local (purely differential) Hamiltonian operators (with corresponding functionals), a property which is not shared by all integrable systems! Two well known, simple examples are the NLS and MKdV equations.

A student reading this book would not be inspired to pursue the subject any further. There is no general framework developed and there are no general techniques presented. After reading this book the student would not be able to use any of these ideas to study a new equation or physical system. My advice to the student would be to avoid this book and turn to the above listed references. My advice to the publishers is to change referees. The book should never have been published.

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks Chapman & Hall for providing a review copy of this book.


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