Bibliography

L. Landau and L. Pyatigorsky. Mechanics. (Theoretical Physics under the general editorship of Prof. L. D. Landau, vol. I.) Gostekhizdat. Moscow—Leningrad, 1940, 200 pp., price 7 rubles.

The book under review* is the first part of a five-volume course of theoretical physics planned by Professor Landau. In the introduction it is therefore preceded by a definition of the subject of theoretical physics. This definition contains a number of debatable propositions. Some of them deserve to be noted, since they have affected the entire exposition.

The authors point out that not only the establishment of general laws, but “even the finding of consequences from general laws must be preceded by an experimental study of the phenomena.” This proposition is correct to a certain extent, for, as the authors rightly say, “without such a study it is often impossible to determine which of the enormous number of factors involved are essential and which may be neglected.” But it also contains the danger of an error consisting in adjusting the consequences of general laws to a result already known in advance from experiment. A whole series of examples of such an error can be found in the book under review.

In the authors’ opinion, theoretical physics should have an exclusively qualitative character, whereas determining the numerical values of physical quantities, generally speaking, is not among its tasks. It is difficult to agree with this proposition, since without the ability to determine the numerical values of physical quantities one cannot speak of verifying general physical laws, which, according to the authors themselves, manifest themselves in the form of dependencies between physical quantities, i.e. between their numerical values.

The authors consider mathematical rigor not only unnecessary, but even very harmful. They assert that “overly precise calculations ... may even lead to the regularities existing in a given phenomenon falling out of consideration altogether.” How this can happen is completely incomprehensible to us. If, upon being brought to such a conclusion, “the regularities have fallen out of consideration,” then, in our opinion, what should be blamed is not mathematical rigor, but an incorrect formulation of the problem. This is so when a certain mathematical rigor is observed. In the absence of mathematical rigor, regularities can indeed (and quite easily) fall out of consideration altogether.

The authors’ negative attitude toward mathematical rigor apparently extends to rigor in reasoning in general. In any case, the book abounds in examples of non-rigorous reasoning. Some of them also lead to incorrect conclusions. If the number of such incorrect conclusions in the book is relatively small, this must be attributed simply to the fact that the desired result is known to the authors in advance. Such a view of rigor in reasoning leads, however, to the reader’s being left unclear as to what follows from what and

* This review was received by the editors in July 1941. Its publication was delayed because of the interruption in the journal’s appearance.

why. Meanwhile, the main purpose of any textbook, in our opinion, should consist precisely in showing the reader the logical connection between the concepts discussed in it.

Let us proceed to an analysis of the main content of the book.

The first two chapters contain (or, in the authors’ intention, are supposed to contain) an exposition of the basic principles of mechanics; the remaining chapters are applications of them to individual problems.

In the first chapter, one should note above all the absence of a definition of the subject of mechanics. On p. 13 one encounters the assertion: “Hamilton’s principle is a law of motion of every mechanical system.” This assertion is incorrect, since there are systems that are nonholonomic and dissipative (with friction). They are mentioned, it is true, very briefly, in this book (§§ 34 and 49). In any case, the consideration of such systems should be included in the subject of mechanics.

There is also no explanation of the basic mechanical concepts, including the concepts of force and mass. The enormous methodological advantage of mechanics—its visual clarity—is not used. Therefore the first chapter of the book can be understood only by a person who already knows mechanics.

The construction of mechanics is based on the principle of least action (Hamilton’s principle). The authors proceed here from the erroneous notion that “under given external conditions the motion is completely determined by the coordinates of the beginning and end of the motion” (p. 152). That this notion is incorrect is especially clear from the example of the free motion of a material point on the surface of a sphere. If the north and south poles, the meridian along which it moves, are taken as the initial and final points, it remains undetermined until the direction of its initial velocity is specified.

In fact, when the Lagrange equations are given, the motion is determined by the initial coordinates and initial velocities. And since the specification of these latter is not equivalent to the specification of the final coordinates, one cannot speak of equivalence between Hamilton’s principle, on the one hand, and the Lagrange equations with initial conditions, on the other.

For this reason, to make the principle of least action the basis of mechanics is hardly correct, even apart from the fact that this principle is applicable not to all systems. We are not saying that the principle of least action is more difficult than the equations of motion and that, in our opinion, one should begin with the easier; in questions of method different opinions are possible.

Speaking of the principle of least action, we had in mind the well-known extremal property of the action integral: the vanishing of its first variation. The authors, however, understand this name literally: they believe that the action integral always “has a minimum value for the actual motion” (p. 13). That this is incorrect is shown by the same example of the motion of a point on a sphere. If the initial and final positions are not poles, then “straight” motion and “circumnavigating” motion are possible. For straight motion the action integral has a minimum, while for circumnavigating motion it does not. In the general case one may only assert that the action integral has a stationary value in the sense that its first variation is equal to zero.

One should condemn the authors’ tendency to derive everything, even obvious things, from far from obvious general principles, and moreover in a non-rigorous way. A typical example is the following. The authors do not give a physical definition of mass from which it would follow that it is always positive. Mass is defined by the authors as a proportionality factor in the Lagrange function of a free material point. It is clear that exactly nothing can follow from such a definition, since this factor can simply be reduced. Meanwhile, the authors believe that on the basis of such a definition one can prove the positivity of mass from the principle of least action. In doing so, the authors understand the principle of least action

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literally, i.e. incorrectly, and this understanding is essential for them. We readily believe that mass is positive, but we can in no way agree that this follows from their reasoning.

The arguments preceding the introduction of an inertial coordinate system are incomprehensible, and the definition of such a system is hardly correct, as “motionlessly connected with certain freely moving bodies” (p. 17). Under this definition there would fall a system connected with a freely flying rotating projectile.

On p. 22 it is said: “The Lagrange function possesses the very important property of additivity.” But immediately a formula is given from which it follows that it does not possess this property, for it contains the mutual potential energy of particles, which is not additive. Thus it remains unknown whether, in the end, the Lagrange function possesses this important property or not.

The concept of force is introduced only in § 8, and forces depending on velocities are not considered at first. Thus not only dissipative forces, for which the Lagrange function does not exist, but also gyroscopic and magnetic forces fall outside the discussion. When these forces, as well as forces arising from reactions of constraints (holonomic and nonholonomic), are finally subjected to consideration (§ 34, § 49, etc.), their connection with the initial definition of force given in § 8 remains unclear.

One could also object to the use of the term “impulse” in the sense of “quantity of motion.” Mechanics is not a new science, and a definite terminology has become established in the Russian literature. By “impulse of a force,” or simply “impulse,” it is customary to understand the integral of the force over time, and by “quantity of motion” the product of the mass and the velocity. The increment of the quantity of motion is equal to the impulse of the force, but these two concepts should not be confused (just as one should not confuse the concepts of force and the product of mass by the acceleration to which it is equal).

In the chapter on small oscillations, the system of linear equations for the amplitudes is treated incorrectly (p. 83). There is no necessary, in our opinion, and moreover very simple proof of the reality of the roots of the characteristic equation (frequencies). The case of multiple roots, extremely important in practice, is not only not considered, but there is not even any mention that such a case is possible. In connection with this there is no indication that only because of the special form of the equations of mechanics are secular terms not obtained in the case of multiple roots. The solution of the system of equations, and also the normal coordinates, are expressed through the minors of the determinant; in the case of multiple roots, however, all these minors, as is known, are equal to zero.

The method of successive approximations in the case of anharmonic oscillations (§ 33) is expounded unsatisfactorily. The authors’ method of calculation does not give the dependence of the period on the amplitude and leads to secular terms in the expression for the coordinates.

Much of what relates to the case of dissipative forces (§ 34) is expounded unsatisfactorily. It is unclear what considerations, apart from mathematical simplicity, compelled the authors to restrict themselves to the case of a linear dependence of friction forces on velocity. If a linear dependence is adopted, then, contrary to the authors’ assertion, the linear terms in the equations of motion are entirely arbitrary: by separating them into gyroscopic (antisymmetric) and dissipative (symmetric) ones, we always obtain a definite dissipative function. Statistical physics is absolutely irrelevant here, and the reference to it on p. 97 is incomprehensible.

On p. 106 the equations of oscillations with damping and their characteristic determinant are set up. Its roots are not investigated; of them only the following is said: “From physical considerations it is clear that the real parts of these roots will be... negative.” This seems to us completely insuffi-

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it permissible to refer to physical considerations in the process of solving a problem which has already been completely formulated mathematically. Moreover, such a reference does not relieve one of the necessity of mathematical investigation: if the mentioned physical considerations are regarded as obvious, then, in order to avoid illogicality, a special proof is required that the adopted formulation of the problem does not contradict them. Such illogicality may perhaps be permissible in especially complicated problems, where the mathematical investigation is difficult, but not in the problem of small oscillations, where everything turns out to be quite simple.

Chapter V, on the motion of a rigid body, apparently contains no particular inaccuracies. By contrast, Chapter VI, on canonical equations, contains a number of major blunders.

What is said at the beginning of § 52 (p. 141) about generating functions is incomprehensible (and unnecessary). It is also unclear on which variables they depend. We have already mentioned the error in § 55 (p. 152) in our discussion of the authors’ treatment of the principle of least action. The derivation in § 56 (p. 155) of Hamilton’s equations from the variational principle is incorrect: the variations \(\delta q\) and \(\delta p\) are not independent. § 58 begins with an attempt to consider the general integral of the Hamilton–Jacobi equation (which is not at all needed for the mechanics problem). This attempt is not carried through to the end: on the very next page (p. 163) we read: “We shall not prove that this is a general, and not a particular, integral.” § 59 (p. 164) begins with the paradoxical assertion: “There is no general method for integrating the Hamilton–Jacobi equation.” What, then, are the Cauchy method, the first and second Jacobi methods, and other known methods? If by integration the authors mean integration in finite form, then their assertion becomes trivial.

§ 62 is devoted mainly to the classification of types of motion of mechanical systems. This paragraph deserves special consideration, since it contains an especially large number of errors. The arguments here lay claim to generality, which, however, they do not possess. In many cases one cannot even grasp the train of thought. Most of the assertions are dubious, and some can be refuted by simple examples. Let us try to trace some of them.

On p. 178 there is an attempt to prove an obviously false assertion: if the coordinate \(q\) tends to a finite limit (limiting motion), then the corresponding momentum \(p\) must increase without bound. This assertion can be refuted by the example of a mathematical pendulum moving from a position of unstable equilibrium: in this example the momentum tends to zero, not to infinity. Let us now see what the “proof” of this assertion consists in.

The starting point is the equalities:

\[ p=\frac{\partial S_0(q,\alpha)}{\partial q};\qquad \beta=\frac{\partial S_0(q,\alpha)}{\partial \alpha}, \tag{1} \]

whence

\[ \frac{\partial \beta}{\partial q}=\frac{\partial p}{\partial \alpha}. \tag{2} \]

The authors then reason as follows:

“Since, as \(t\to\infty\), the variable \(\beta\) tends to \(+\infty\) or to \(-\infty\), while the variable \(q\) tends to a finite limit, we may conclude that

\[ \frac{\partial \beta}{\partial q}\to \pm\infty . \]
”

This conclusion is refuted by the example

\[ q(\beta)=\frac{\sin(a\beta^2)}{\beta},\quad \beta=\omega t. \tag{3} \]

Here, as \(t \to \infty\), \(q \to 0\), whereas \(dq/d\beta\) does not tend to any limit.

But let us look further.

“On the basis of the preceding equality (2) we see that in this case also \(\dfrac{\partial p}{\partial a}\to \pm\infty\). It can easily be shown that not only \(\dfrac{\partial p}{\partial a}\), but the impulse \(p\) itself must increase without bound, i.e. \(\lim_{t\to\infty}p=\pm\infty\).”

But what if, for example, one takes

\[ p=\sqrt{a-q};\quad q\to a? \tag{4} \]

For such a dependence between \(p\) and \(q\) there will be

\[ \frac{\partial p}{\partial a}\to\infty;\quad p\to 0. \]

Thus, in the argument cited, both premises are false. It is not surprising that the conclusion is also false.

On the same p. 178 the quantity \(\beta\) is assumed to be expressed through \(p\) and \(q\):

\[ \beta=\beta(q,p), \tag{5} \]

after which it is said:

“In the case of conditionally periodic motion, on the right-hand side of the last equality the coordinate \(q\) and the impulse \(p\) tend neither to a finite nor to an infinite limit. The left-hand side of the equality, i.e. the quantity \(\beta\), tends to \(\infty\) or \(-\infty\). Obviously, this is possible only in the case where the variable \(\beta\) is a multivalued function of the variables \(q\) and \(p\).”

We can only add that the obvious absurdity of this conclusion is apparent to us, but by no means its validity.

We have not yet reached the end of p. 178. Let us try to write more briefly and note only the main point. The authors regard as proved that the action function \(S_0\), expressed in terms of \(p\) and \(q\), will be a multivalued function, and consider its change \(\Delta S_0\) in traversing a certain closed contour in phase space (the space \(p,q\)). Then they write the following astonishing equality:

\[ \Delta p=\Delta\frac{\partial S_0}{\partial q} =\frac{\partial}{\partial q}\Delta S_0. \tag{6} \]

Each term of this equality is perplexing. First of all, \(S_0\) is considered here (at least, this is all that has been considered) as a function of \(q\) and of \(p\) (and not of \(q\) and of \(a\)); therefore already it will not be \(p=\dfrac{\partial S_0}{\partial q}\). But this is a trifle. The most important point is that, by definition, \(\Delta S_0\) can depend only on the contour along which the circuit was made, and in no way can it depend on \(q\), or on \(p\), or on \(a\). Therefore the interchange of the signs \(\Delta\) and \(\dfrac{\partial}{\partial q}\) in (6) is not only illegitimate, but in general devoid of any meaning.

...meaning, just like the right-hand side of formula (6). The authors, however, conclude from (6) and from $\Delta p = 0$ that

\[ \Delta S_0 = \Delta S_0(\alpha), \tag{7} \]

i.e., that $\Delta S_0$ is a function of $\alpha$. The conclusion is absurd, since, by the definition given by the authors, $\Delta S_0$ cannot be a function of $\alpha$. All this is written as though the authors did not understand certain very simple things; perhaps they are confusing the phase space $(q,p)$ with the configuration space $(q$ for given $\alpha)$. Suppose, however, that some quantity $\Delta S_0$ is a function of $\alpha$!

We read further (p. 179):

“Of course, in traversing some phase lines $\Delta S_0$ may be equal to zero. The smallest value of $\Delta S_0$ different from zero is called the period corresponding to the function $S_0$.”

But whence does it follow that such a smallest value of $\Delta S_0$ different from zero exists? Could it not be that $\Delta S_0$, taken along a large contour, will be large, taken along a small contour—small, and, with an unlimited decrease of the contour, will tend continuously to zero? After all, the representations borrowed from the theory of integrals of meromorphic functions of a complex variable may be wholly inapplicable here.

However, let us forgive them this sin as well and try to read on.

“In the general case, when a system performing conditionally periodic motion has $n$ coordinates, the function $S_0$ depends on $n$ variables. Therefore there correspond to it $n$, generally speaking, independent periods.”

But whence comes this conclusion? Why do the periods of a system in general have a finite basis? Why are there not only a finite number of them, but precisely $n$ for a system with $n$ degrees of freedom?

It is impossible to answer these bewildering questions, but one may try to answer a question of a psychological order: why did the authors get so confused at this point? We venture to put forward the following hypothesis: the authors became confused here because they transferred to the general case certain results proved for systems with complete separation of variables (these systems are mentioned at the end of § 62).

This hypothesis is confirmed by the fact that on p. 180 the case of one degree of freedom is considered, and immediately after it the general case is discussed.

Be that as it may, what we have here is an incredible confusion of concepts. Let us try to put them in order. On p. 178 the following definition of conditionally periodic systems is given:

1. “In the case ... when the coordinate $q$ does not tend to any limit, the motion has a quasistationary character and is called ‘conditionally periodic’.”

Thus, the single circumstance that the coordinates of the system do not tend to any limit is, in the authors’ opinion, sufficient for the conclusion that all the properties of the systems under consideration hold, including the above-mentioned property of having precisely $n$ independent periods.

This is confirmed by the fact that on p. 181 it is said:

1. “However, it can be shown that... the system returns arbitrarily close to each of its states.”

The indicated property of the system is commonly called stability in the sense of Poisson. One has to state that, in the authors’ opinion, stability in the sense of Poisson follows, in the final analysis, from definition 1. But that is not all. Quotation 2 is immediately followed by the following:

1. “As a consequence of this, the motion is called almost periodic...”

The construction is as follows: the authors consider it obvious and not requiring proof that if a system is stable in the sense of Poisson, then its coordinates are almost periodic functions of time.

But even this is not all. Let \(F\) be some (single-valued) function of \(q\) and \(p\). On the same p. 181, somewhat above, we read:

1. “In the general case, when the system has \(n\) degrees of freedom, the function \(F(w_1 \ldots w_n)\) can also be expanded in a Fourier series.”

Since \(w_i\) is proportional to the time, this is equivalent to the assertion that \(F\), as a function of time, has exactly \(n\) independent periods.

Thus, we find the authors in the following sequence of assertions:

1. The coordinates of the system do not tend to any limit.
2. The system is stable in Poisson’s sense.
3. The coordinates are almost-periodic functions of time.
4. The coordinates are expandable in multiple Fourier series.

All these assertions are considered by the authors to be either equivalent or, at least, consequences of 1. Meanwhile, in reality this is by no means so. Property 1 is a very general property from which it is difficult to infer anything, except in the case of one degree of freedom. Stability in Poisson’s sense is already a much more concrete property, which does not follow from 1. It, in its turn, is much broader than property 3, which is possessed by a relatively narrow class of systems. An even narrower class possesses property 4—expandability in a Fourier series.

If, however, one starts from 4, then 3, 2, and 1 can indeed be derived from it.

Thus, in the § 62 under discussion in Landau and Pyatigorsky’s book there is an unimaginable confusion of concepts. It is very difficult to read this paragraph. At the first attempt to follow the authors’ reasoning one quickly becomes convinced that nothing can be understood.

One cannot but be surprised that such a major scholar as one of the coauthors—Prof. Landau—undoubtedly is could have written a book with so great a number of gross errors.

A certain carelessness of exposition and lack of rigor in reasoning may perhaps still be excusable in new, little-studied areas of physics. But in such an old, established field as classical mechanics, and all the more in a textbook, everything must be impeccably clear and rigorous. This can be achieved by very simple means.

Proceeding to the assessment of the book as a whole, we must admit that the authors have not succeeded. Of course, a whole series of questions is presented correctly in the book. § 17, where the period of motion with one degree of freedom is considered as a function of the energy, is interesting, as are §§ 21–23, devoted to the scattering of particles (where, however, the statistical aspect of the problem is presented at first in too brief a manner). The very attempt to give an exposition of mechanics as a chapter of theoretical physics is interesting, and the selection of material is fairly successful (with the exception of the problems, which have the character of exercises in differentiation). But these positive aspects are far from sufficient for the textbook to be recognized as good. After all, we cannot seriously credit the authors with correctly solving, for example, Kepler’s problem. We have the right to demand much more of them. And in the more subtle and difficult questions of mechanics—the variational principle, the classification of types of motion of mechanical systems—the authors prove decidedly not up to the mark: in the corresponding chapters we find errors and confusion.

It seems to us, however, that Landau and Pyatigorsky’s book can still be corrected. For this, in addition to correcting the errors and lack of rigor, it is desirable, in our opinion, to rework the book also in the direction of having the principle of least action preceded, at least, by a definition of the basic concepts of mechanics. As for the ill-fated § 62, here the simplest thing is to confine oneself to the case of complete separation of variables, where the whole investigation can be carried out explicitly.

We hope to see the book in a second edition corrected and substantially reworked.

V. Fok