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Electromagnetism of continuous media. Mathematical modelling and applications. (English) Zbl 1027.78001

Oxford: Oxford University Press. xvii, 668 p. (2003).
The object of this book, according to the authors’ preface, is to consider the mathematical modeling of electromagnetic solids, including treatments of nonlinearity and memory effects. The book is substantial and consequently it is best to restrict oneself to a brief list of the most interesting contents, reserving comments for the end.
The first section of the book is entitled ‘Basic Notions in Electromagnetism’ and has four chapters. The first chapter is on Electromagnetic fields. This is concerned largely with classical electromagnetic theory arising out of Maxwell’s equations. Included is a treatment of the mechanical properties of an electromagnetic field and a short introduction to the mechanics of deformable media. The second chapter, Green’s functions and retarded potentials, gives a highly abstract treatment of Green’s functions and goes on to consider other topics such as the telegraph equation and the initial value wave equation. There is also a considerable amount of the classical theory such as Kirchhoff’s solution. The third chapter, Time-harmonic fields contains such elementary topics as the Helmholtz theorem and plane waves. There is also a discussion of group velocity, signal velocity and dispersion. Other subjects mentioned are the Doppler effect, and principles of reciprocity and stationary phase. The fourth chapter, Models of materials with memory, introduces what might be termed unorthodox media. Amongst the materials considered are ferrites, ferroelectric media, certain metals, the ionosphere and plasmas. There is a brief introduction to the principle of causality and there are discussions of the Kramers-Konig relations and the Clausius-Duhem inequalities.
The second section is entitled ‘Thermodynamics and Mathematical Problems’. The fifth chapter, Thermodynamics of simple mathematical systems, considers in detail the problem of media with memory and enthalpy. The governing thermodynamic laws together with the ideas of reversibility and controllability are considered and the topological properties of the abstract history spaces associated with dielectrics and conducting media with memories are presented. Problems associated with interfaces are indicated. The sixth chapter, Thermomagnetic systems, is a short one. Some consequences of the second law of thermodynamics are introduced, there is a discussion of the relation between controllability and entropy together with a comment on the concept of pseudo-energy and a development of the Onsager reciprocal relations. The seventh chapter, Existence and uniqueness, is somewhat miscellaneous in character. It contains introductions to a number of topics such as rector analysis, resonators and strong and weak solutions of differential equations. Other topics are absorbing boundaries and spatial decay on thermoelectromagnetic conductors. The contents of the eighth chapter, Wave propagation, are much what one might expect. The classical theory of plane waves and polarization is discussed to begin with, particular attention being paid to dissipative media. This leads on to magnetohydrodynamic waves together with waves in anisotropic and gyrotropic media and media with memory. The ninth chapter, Extremum principles, contains some appropriate applications of the Calculus of Variations to various types of electromagnetic media. Rayleigh’s principle is mentioned in connection with resonators and there is a discussion of the modification of Maxwell’s equations to allow for photons of non-zero mass. Principles associated with some nonlinear materials are mentioned and Reiss-type principles are briefly referred to.
The third part entitled ‘Nonlinearity and nonlocality’ is possibly the most important part of the book. The tenth chapter, problems in nonlinear electromagnetism, opens with treatments of nonlinear optics and of nonlinear constitutive equations with memory. The Luxembourg effect is explained and the concepts of single waves and shocks are introduced. The chapter culminates in a treatment of shocks with memory. The eleventh chapter, nonlocal electromagnetism and superconductivity, opens with a discussion of entropy and the second law of thermodynamics. There is a further treatment of nonlinear dielectrics. There is a description of the phenomenon of superconductivity and descriptions are given of a number of theories supporting to explain it. In the twelfth chapter, magnetic hysteresis, there is a brief discussion of the phenomenon and of the ideas of micromagnetics. The difficulties of a formal mathematical analysis are mentioned and rate-type, Preisach, bilinear Coleman-Hodgon models are described. History-rate and history-integral life models are also touched on.
The main portion of the book is followed by a number of appendices and a variety of topics – Bessel functions, Fourier transforms, Sobolev spaces, compact operators, eigen functions, curvilinear coordinates, finite electromagnetics. There is a list of 192 references (25 in Italian, 5 in German and 5 in French), one or two of which have not yet been published, and a satisfactory index.
Considering the book as a whole it may be regarded as a worthy continuation, so to speak, of well-known works on electromagnetic theory such as Stratton and Smythe, although the notation is not always quite the same. The title is in fact slightly misleading, there being in fact a lot of thermodynamics in the book. I feel that there are one or two minor faults in it. There could be more diagrams, in some cases equations are dimensionally incorrect e.g. 7.51 where two terms of different dimensionality have been added, and occasionally proofs are not given, the reader being referred to one of the references. These is very little on topics such as waveguides and diffraction and no mention of Schwinger’s variational principle which was used with great effect in the 1940’s. These are small faults, however.
The book is well written and the printing as might be expected from this publisher is excellent. It is not a book to be rushed through, but requires solid reading. It can be thoroughly recommended. Unfortunately, the high price will put it out of reach of nearly all individuals, and many UK University libraries.

MSC:

78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A60 Lasers, masers, optical bistability, nonlinear optics
80A17 Thermodynamics of continua
82D55 Statistical mechanics of superconductors
49S05 Variational principles of physics
74F15 Electromagnetic effects in solid mechanics
74A15 Thermodynamics in solid mechanics
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