`UK Nonlinear News`,
`August 2000`

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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