L. Landau and E. Lifshitz. Continuum Mechanics. (Theoretical Physics, edited by Prof. L. D. Landau.) Gostekhizdat. Moscow—Leningrad, 1944, 624 pp., price 33 rubles.

The book under review is the third volume of the course in theoretical physics edited by Prof. L. D. Landau, and is devoted to the exposition of hydrodynamics, the theory of elasticity, and certain related questions.

The presentation of hydrodynamics in courses of theoretical physics is usually very compressed and, in any case, covers only the classical range of questions, which does not include turbulence, the theory of the boundary layer, the theory of explosions, etc. The situation is analogous, although somewhat more favorable, with respect to the theory of elasticity.

What has been said above concerning courses in theoretical physics applies no less to the teaching of this discipline in universities: in the general plan of instruction in theoretical physics at the physics faculties, continuum mechanics has, at least until the very recent past, occupied a quite insignificant place or has even not been represented at all.

To a certain extent this situation is connected with the tradition established among us of assigning continuum mechanics entirely to mathematical physics; the latter has found reflection also in the fact that specialization in the field of hydro- and aerodynamics and the theory of elasticity is carried out at the mechanics-and-mathematics faculties, not at the physics faculties of universities. Meanwhile, the entire development of hydrodynamics and adjacent fields in recent times convincingly shows that the leading methods here are precisely those of theoretical physics, i.e., an approximate treatment of questions connected with penetration into their physical essence, and not a rigorous solution of long-known equations of the Navier–Stokes type. Therefore there appears to be an absolutely unquestionable necessity to break down the Chinese wall between theoretical physicists and hydrodynamicists, and consciously and broadly to attract the methods of theoretical physics to the solution of hydrodynamic problems.

Recently, in particular in connection with the demands of wartime, the situation in this respect has improved considerably, which has immediately affected practice—it is enough to mention a number of works devoted to hydrodynamics and related questions by L. D. Landau, Ya. B. Zel’dovich, V. G. Levich, S. Z. Belen’kii, and others. But the greatest role, from the point of view of including continuum mechanics in all its breadth in the system of theoretical physics, is called upon to be played precisely by the splendid book of L. Landau and E. Lifshitz considered here.

The authors, decisively breaking with tradition, have included in their course of theoretical physics almost all the basic questions of modern continuum mechanics, not limiting themselves only to its fundamental propositions and classical problems. In order to be convinced of this, it is enough at least briefly to become acquainted with the contents of the book.

In its first two chapters the basic propositions of the hydrodynamics of an ideal and of a viscous fluid are set forth. The exposition here is everywhere concise, but at the same time very precise and clear; moreover, the derivation and discussion of the basic equations and relations are not cluttered with a forest of various secondary problems and transformations, or with proofs of theorems of vector analysis, as is often observed in courses on hydrodynamics. What has been said, however, applies to the whole book as a whole.

In these same chapters the solution of a number of important classical problems is carried out, many of which are placed in petit type precisely as problems to the corresponding paragraphs of the book; this device contributes greatly to the harmonious structure of the whole exposition and makes it possible to avoid the appearance of loose, lengthy paragraphs. Among the problems mentioned we shall point to the consideration of gravitational

waves, the determination of the drag force in potential flow, the derivation of the Lualey and Stokes formulas, the investigation of certain exact solutions of the Navier–Stokes equations, etc.

The third chapter is devoted to turbulence. It may be said without any exaggeration that in this chapter, for the first time, a relatively coherent (and accessible to understanding by persons acquainted only with hydrodynamics) exposition of this most confused question is presented. This is explained to a considerable extent by the fact that the book sets forth the results of the very latest works in the field of turbulence (A. N. Kolmogorov, L. D. Landau, and others), which have not yet been covered in the survey literature.

Chapter IV gives a detailed account of boundary-layer theory, and Chapter V of the theory of heat conduction, heat transfer, and convection in fluids.

Chapter V sets forth acoustics, i.e. the theory of the emission, propagation, absorption, and scattering of sound waves. The entire exposition is, of course, conducted on the basis of exact hydrodynamic equations, simplified in accordance with the character of the problem (in the present case, with the propagation of small oscillations in a liquid). Therefore all the general formulas (for example, expressions for the density of sound energy and sound pressure, the reciprocity theorem, etc.) are obtained in such a way that no doubt can arise as to their validity, as happens when acoustics is expounded from the very beginning on the basis of linearized equations.

Chapter VII is devoted to a rather complete theory of discontinuous solutions of the equations of hydrodynamics. With regard to this chapter one may, to a considerable extent, repeat what was said about the chapter devoted to turbulence.

Chapter VIII sets forth the theory of the flow of bodies by a stream of compressible fluid and, in particular, the theory of flow in supersonic motion. Further, Chapters IX–XI consider the hydrodynamics of combustion (slow combustion, detonation, propagation of a detonation wave, etc.), diffusion and thermodiffusion, and surface phenomena (Laplace’s formula, capillary waves, and other questions).

The first part of the book, devoted to hydrodynamics and occupying more than two thirds of its volume, ends with a comparatively short Chapter XII, in which the kinetic theory of gases is set forth. The exposition here seems to us less successful than in the other chapters, primarily because of the almost complete absence of examples of concrete solutions of the equations obtained. At the same time, the derivation of the kinetic equation and its discussion, not on a model of elastic spheres but in a general form, deserve attention and are very valuable.

The second part of the book sets forth the theory of elasticity, and its first chapter (i.e. Chapter XIII of the book) is devoted to the symmetry of crystals. This latter question, i.e. the systematic consideration of crystal symmetry and the classification of crystals, usually drops out of courses in theoretical physics, something that can only be regretted, since without acquaintance with the symmetry of crystals it is impossible to consider problems of crystal physics.

In Chapter XIV the basic equations of elastic equilibrium are derived and discussed; here too some problems belonging to this subject are solved (equilibrium of an elastic medium bounded by a half-plane, problems on the contact of elastic bodies), and elastic properties of crystals are briefly considered. Next the equilibrium of rods, plates, and shells is considered (Chapter XV), motion in an elastic medium, i.e. elastic waves (Chapter XVI), and finally heat conduction and viscosity in solids (the last chapter—Chapter XVII).

From the far from complete list given above of the questions contained in the book, one must think that the breadth of the coverage of the material relating to the mechanics of continuous media is sufficiently clear. For this the authors have had to pay something: in a number of cases, as it seems to us, the exposition is too compressed, and its, in general, positive compactness begins to hinder understanding. The same applies to the very frequent absence of reservations concerning the limited applica-

of the validity of the propositions stated and of the conditions under which they are not satisfied. For example, the derivation of the most important expression for the stress tensor on pp. 46–47 occupies only a few lines, and questions of uniqueness, and especially the limits of applicability, remain blurred. The very important proposition that the force on an element of a surface is equal to the flux of momentum through this element is given without any explanation (p. 49). In deriving the kinetic equation in § 100, the absence of external forces is assumed tacitly and, moreover, so abruptly that this circumstance loses its importance. On p. 495 the symmetry of the stress tensor is proved in such a way that the impression may be created of its inevitability. Yet, as is known, in the presence of electric forces the stress tensor is not symmetric. And, in general, the authors are not particularly fond of specifying the conditions for the validity of their conclusions and, it seems to us, rather overuse arguments of the type “obviously,” “from this it is clear,” “consequently,” etc., in cases where even to a qualified reader the inevitability of the corresponding conclusion is not at all clear.

The conciseness of the exposition has made itself felt especially strongly in Ch. XIII, devoted to the symmetry of crystals. This very difficult question to master needs a significantly larger number of examples, illustrations, and so forth.

One would very much like, in the same book in which the question of symmetry is discussed, also to see an exposition of the basic problems of crystal physics (piezoelectricity, pyroelectricity, etc.). All the more so since in most cases they are connected with the theory of the elasticity of crystals, which is touched upon too briefly in Ch. XIV.

It must, however, be borne in mind that if the authors can easily correct individual rough spots, supplementing the book with new material or substantially detailing the exposition of material already present involves well-known difficulties: the simplest and best way to resolve them is probably to take the path of increasing the size, dividing the book in subsequent editions into two half-volumes, i.e. into hydrodynamics and the theory of elasticity.

Here it would be inadvisable to dwell on minor shortcomings or misprints found in the book, although some of them are quite annoying (for example, on p. 479, instead of a center of symmetry there appears a plane of symmetry, etc.).

We shall therefore confine ourselves to just one comment. The authors give absolutely no literature references; yet, by referring to articles in which one or another of the problems under discussion has been solved, they could both save space, avoiding a number of details, and, what is most important, help the reader, if necessary, to become acquainted with the original literature.

The foregoing remarks cannot, of course, affect the overall assessment of the book. This assessment is the highest. The book by L. Landau and E. Lifshitz is the first work in world literature in which contemporary hydrodynamics, in all its richness and variety, is presented as a part of theoretical physics.

At the same time, the great coherence and clarity of the exposition, together with its almost encyclopedic completeness and high scientific level, make this book valuable for a very wide circle of readers—from students to specialists in theoretical physics and hydrodynamicists inclusive. Reviews of good foreign books usually end with a recommendation that they be translated into Russian. In the present case we would like to make a proposal of the opposite kind, namely to recommend that care be taken to translate the book by L. Landau and E. Lifshitz into English, in order to acquaint foreign scholars with the finest new work in our scientific literature.

V. L. Ginzburg