The books once interested me

coolboy

周华 发表于 2017-5-17 21:45
Dear Dr. Coolboy,

Thank you for your photos and letters posted here in this forum.

It was my pleasure and thank you for your effort in managing a nice professional website.

coolboy

I mentioned above that I first learnt the interesting features of Dirac-delta function while reading the book by B. Friedman when I was at high school. Later, when I was at college, I accidently re-learnt the Dirac-delta function as a special example of the so-called generalized functions from a mathematical perspective. On the last day of 1980, I bought the first English professional book in my life, which was a volume-1 textbook by Richtmyer (1978) on advanced mathematical physics. I still kept the receipt in the book. It was a photocopy book (影印书) and, at the time, I was asked to show my student identification in order to get into the bookstore and to make a purchase. The cost to the book (2.40 Yuan in RMB) was much less than the price for the original copy but it still cost me a big fortune for the average income (~30 Yuan/month) at the time was very low. The first few chapters of the book by Richtmyer (1978) presented a systematic development of the generalized functions including the Dirac-delta function. Another classic and popular textbook on this subject was by Lighthill (1958).


Lighthill, M. J., 1958: Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, London and New York, 79 pp.
Richtmyer, R. D., 1978: Principles of Advanced Mathematical Physics. Vol. I. Springer-Verlag, New York, 422 pp.
Richtmyer, R. D., 1981: Principles of Advanced Mathematical Physics. Vol. II. Springer-Verlag, New York, 322 pp.

coolboy

Richtmyer’s monograph on the subject of mathematical physics was at an “advanced” level. A classic monograph at an “intermediate” or a “normal” level on the same subject was the one by Courant and Hilbert:

Courant, R., and D. Hilbert, 1953: Methods of Mathematical Physics. Vol. 1. Wiley-Interscience, New York, 560 pp.
Courant, R., and D. Hilbert, 1962: Methods of Mathematical Physics. Vol. 2. Wiley-Interscience, New York, 831 pp.

Richtmyer argued that the materials presented in the above monograph by Courant and Hilber were mostly established in the 19th century whereas the two major advances of quantum mechanics and general relativity in the 20th century physics required additional mathematical ideas and tools that had been well established in the 20th century. The major and systematic fresh contents included in Richtmyer’s monograph that were not included in other books in this subject were “real analysis” (including functional analysis, especially important to establish a rigorous framework for probability theory), “group theory” (important for quantum mechanics, spectroscopy theory, fluid mechanics, etc), and “manifolds” (important for general relativity, fluid mechanics, etc.).

It appears that both the monograph by Courant and Hilbert and the one by Richtmyer were for students majoring in applied mathematics and theoretical physics. For students majoring in general physics, chemistry, and engineering, two popular textbooks on the subject had been:

Arfken, G., 1985: Mathematical Methods for Physicists. Third Edition. Academic Press, Inc., New York, 985 pp.
Gustafson, K. E., 1999: Introduction to Partial Differential Equations and Hilbert Space Methods. Third Edition. Dover Pub. Inc., New York, 448 pp.

coolboy

R. D. Richtmyer was much better known for the following classic book in the field of computational physics:

Richtmyer, R. D., and Morton, K. W., 1967: Difference Methods for Initial Value Problems. Second Edition. Wiley & Sons, Inc., 406 pp.

coolboy

Two good math handbooks that I often use and are highly recommended:

《数学手册》编写组，1979：《数学手册》，人民教育出版社，北京，1398页。
Pearson, C. E. (editor), 1983: Handbook of Applied Mathematics. Selected Results and Methods. Second Edition. Van Nostrand Reinhold Company, New York, 1307 pp.

coolboy

A good reference book in a handbook style on mathematical physics on my bookshelves:

Triebel, H., 1986: Analysis and Mathematical Physics. D. Reidel Pub. Company, Boston, 456 pp.

coolboy

A few specialized handbooks at hand:

Huba, J. D., 2007, 2014: NRL Plasma Formulary. NRL, Washington, DC, 71 pp.
Lodders, K., and B. Fegley, Jr., 1998: The Planetary Scientist’s Companion. Oxford Univ. Press, Oxford, 371 pp.
Mayaud, P. N., 1980: Derivation, Meaning, and Use of Geomagnetic Indices. American Geophysical Union, Washington, DC, 154 pp.
Tohmatsi, T., 1990: Compendium of Aeronomy. Terra Sci. Pub., Tokyo, 509 pp.
Wildi, T., 1995: Metric Units and Conversion Charts. A Metrication Handbook for Engineers, Technologists, and Scientists. Second Edition. IEEE Press, New York, 130 pp.
Woan, G., 2000: The Cambridge Handbook of Physics Formulas. Cambridge Univ. Press, Cambridge, 219 pp.
Zombeck, M. V., 1990: Handbook of Space Astronomy and Astrophysics. Second Edition. Cambridge Univ. Press, Cambridge, 440 pp.

coolboy

I mentioned above that I was quite excited when first learning the Dirac-delta function that had strange features as a function. Later, while reading the book by Richtmyer (1978) on advanced math-physics, I re-learnt the Dirac-delta function from a different perspective and got even more excited. Specifically, I was excited to learn that the traditional one-to-one correspondence between two variables as the concept of function was no longer holding even under the concept of generalized function. For example, if we denote the Dirac-delta function to be [delta](x)=d(x), then the functions d(x)^2, d(x^2), exp[d(x)] were all meaningless even under the extended definition of the generalized functions. Fortunately, in practice, one usually does not encounter this type of functions but only the simple ones such as d(Ax+B) or d(x^2-A^2).

Gaining new insights while learning old materials is one kind of high-level pleasures I sometimes encountered while studying science. Another even more interesting and much more exciting example was about the Riccati equation. I learnt the Riccati equation also when I was in high school. Riccati equation is a nonlinear ordinary differential equation that can be linearized and solved analytically by a transformation of its dependent variable. In general, nonlinear equations cannot be solved analytically (or in a closed form) and the main characters of their solutions thus cannot be easily understood. The fact that the Riccati equation can be solved analytically by a transformation of variable makes it an important example in the field of nonlinear differential equations. It also inspired many scientists to seek different transformations to solve various types of nonlinear equations. Later (1960s-1980s), the concept of solitons or solitary waves became a hot topic after people discovered that a large class of nonlinear partial differential equations can be solved by the so-called inverse scattering method. At the time when I was at college I learnt that a key step to solve the KdV equation, which was the most popular equation on solitary waves, by the inverse scattering method happened to be the application of an old friend of the Riccati equation (e.g., Drazin and Johnson 1989). Realizing the connection between the Riccati equation and KdV equation greatly excited and inspired me at the time.

A much bigger surprise or excitement I received on the old friend Riccati equation came much later when I learnt its very interesting and important usage in applications of turbulence in a particular field....... (to be continued)


Drazin, P. G., 1983: Solitons. Cambridge Univ. Press, 136 pp.
Drazin, P. G., and R. S. Johnson, 1989: Soliton: an Introduction. Cambridge Univ. Press, 226 pp.

I also took a class on solitons taught by Professor P. G. Drazin in mid-1980s. Other well-known books authored by P. G. Drazin were:

Drazin, P. G., and W. H. Reid, 1981: Hydrodynamic Stability. Cambridge Univ. Press, Cambridge, 527 pp.
Drazin, P. G., 1992: Nonlinear Systems. Cambridge Univ. Press, 317 pp.

coolboy

Riccati equation is a first-order nonlinear ordinary differential equation and can be written as

(1)   du(x)/dx + p(x)*u(x) + q(x)*u(x)^2 = r(x)

where x is an independent variable, u is a dependent variable, coefficients p and q are functions of x, and r is an external source term. The equation is nonlinear because it contains a term involving a nonlinear function (u^2) of the dependent variable u. By a transformation of the dependent variable u

(2)   v(x) = [du(x)/dx]/[q(x)*u(x)]

Eq. (1) can be converted into a second-order linear ordinary differential equation with respect to v(x):

(3)   v"(x) + [p(x) – q'(x)/q(x)]v'(x) – r(x)*q(x)*v(x) = 0

where v'(x)=dv(x)/dx and v"(x)=d[dv(x)/dx]/dx denote the first and second derivatives, respectively. Equation (3) is linear and analytically solvable. Given the solution v(x), one is able to integrate Eq. (2) to solve for u(x) for the original Riccati equation. So far, everything is great and well known. To simplify our discussion, we now assume the coefficient q(x)=1. Then, Eqs (1)-(3) can be simplified into
(1b)   du(x)/dx + p(x)*u(x) + u(x)^2 = r(x)
(2b)   v(x) = [du(x)/dx]/u(x)
(3b)   v"(x) + p(x)*v'(x) – r(x)*v(x) = 0

The traditional or the most popular view has always been that Eq (1b) is more difficult to solve than Eq. (3b). The great excitement came when people realized that in some cases, Eq. (3b) is difficult to or even cannot be solved whereas its counterpart of Riccati Eq. (1b) can be solved systematically. One such a special case is when the coefficient r(x) in Eq (3b) is a random function. There are no easy ways to solve differential equations with random coefficients. On the other hand, r(x) appears as an external source in Eq (1b), which can often be easily incorporated into the final solution. Specifically, for a linear operator on the left-hand side, one can express the final solution as a convolution between a Green's function and an external source (e.g., Roach 1982; Stakgold 1979). Under such a circumstance, the statistical property of the solution u(x) is directly tied to that of the external source r(x).

Now, Eq (3b) is one-dimensional. The second derivative v"(x) in one-dimension corresponds to a Laplacian in multi-dimension. As a result, Eq (3b) in two- or three-dimension corresponds to a Helmholtz equation that describes wave propagation in a media. The coefficient r(x) in (3b) is directly related to the refraction index of the media. An important problem in turbulence is to study wave propagation through turbulence and a powerful method to solve this problem is the so-called Rytov's transformation that essentially converts the Helmholtz equation with a random coefficient into an equation with a random external source (e.g., Monin and Yaglom 1975, Ch. 9; Wheelon 2003), corresponding to a reverse thinking （反向思维） of solving the Riccati equation.

Of course, we now need to find a different way to systematically solve a nonlinear equation ......


Roach, G. F., 1982: Green's Functions. Second Edition. Cambridge University Press, New York, 325 pp.
Stakgold, I., 1979: Green's Functions and Boundary Value Problems. John Wiley & Sons, New York, 638 pp.

Monin, A. S., and A. M. Yaglom, 1975: Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. 2. MIT Press, Cambridge, Massachusetts, 874 pp.
Wheelon, A. D., 2003: Electro_m_a_g_n_e_t_i_c Scintillation. II. Weak Scattering. Cambridge Press, Cambridge, 440 pp.

coolboy

If the magnitude of nonlinear terms in an equation is small in comparison with other terms, the equation can often be solved approximately by the so-called perturbation method. By expanding the solution into a series of many terms powered by a small parameter, one may solve a set of linear equations to derive successive approximations of the exact solution. Such an approach is especially suitable to the problems of wave propagation through turbulence medium because the difficult component of the random variation for the medium refraction index is a small parameter (Wheelon 2003).

In some cases, the above approach of the so-called “regular perturbation” method may not work. This happens when the approximate series does not converge. Physically, this often occurs when the introduced nonlinear terms not only change the overall magnitudes of the solution corresponding to the approximate linear equations but also change the fundamental structure of the solution. For example, if both solutions are sinusoidal waves but they have different frequencies, then it is usually an ill-conditioned fit to approximate one wave by a set of the other waves with a fixed frequency and varying amplitudes. To overcome the shortcomings encountered in the “regular perturbation” method, a fascinating research field of the so-called “singular perturbation” methods was systematically and extensively developed in the field of applied mathematics and mechanics in the mid-20th century. I have the following three books that more or less summarized the principles and applications of the singular perturbation methods:

Van Dyke, M., 1975: Perturbation Methods in Fluid Mechanics. Annotated Edition. The Parabolic Press, Stanford, Calif., 271 pp.
Nayfeh, A. H., 1981: Introduction to Perturbation Techniques. John Wiley & Sons, New York, 519 pp.
Kevorkian, J., and J. D. Cole, 1981: Perturbation Methods in Applied Mathematics. Springer-Verlag, New York, 558 pp.

The authors of these books were well-established in this particular field of singular perturbation methods. Among them, van Dyke was best known for his book was originally published in 1964. The so-called “annotated edition”（加注版） was equivalent to an updated or a “second edition” of the book. Instead of completely revising the original edition, the author included a new chapter of “Notes” that listed various progresses occurred in the field since the publication of the first edition. Then, he annotated on the margins of the book on those progresses. Nayfeh was van Dyke's PhD student and his textbook contained many examples, mostly on the ordinary differential equations, which were worked out in detail. Nayfeh's earlier book on the same subject was published in 1973 under a title “Perturbation Methods”. I also have Nayfeh's another book entitled “Nonlinear Oscillations” that was solely devoted to the ordinary differential equations. The book by Kevorkian and Cole (1981) emphasized the multiple-scale technique and applications in partial differential equations. This book was actually a revised and updated version of an earlier edition published in 1968 under the same title by J. Cole. I took a class on perturbation methods taught by Professor J. Kevorkian in mid-1980s. His 1981 textbook was the major reference book for the course. At the time, Kevorkian said that he was preparing a revised/third edition of his book and would be done soon. It appeared that that revised/third edition was published much later in 1996 under a new title “Multiple Scale and Singular Perturbation Methods”.

Though the style and emphasis of the above three books on the singular perturbation methods were different, there was one thing in common. The academic roots of all the authors can all be tracked to a common institute that was especially famous for making significant contributions to this field: California Institute of Technology. In 1940s-1950s, there were quite a few Chinese students and scholars studying or working at the California Institute of Technology. Naturally, many of them would also work in this particular field and made contributions to the field. Specifically, the following four familiar Chinese names also made contributions to this field：林家翘(C. C. Lin)，郭永怀(Y. H. Kuo)，钱伟长(W. Z. Chien)，钱学森(H. S. Tsien).

In the literature on singular perturbation methods, one sometimes sees the term “L-P method” or “PLK method”...... (to be continued)

coolboy

In the literature on singular perturbation methods, one sometimes sees the term “L-P method” or “PLK method”. Here, “L” and “P” in “L-P method” refer to A. Lindstedt and H. Poincare, respectively. On the other hand, “P”, “L” and “K” in “PLK method” refer to H. Poincare, M. J. Lighthill and Y. H. Kuo, respectively. It was sometimes argued that the contribution by Y. H. Kuo was not well justified to associate his name with the method. For example, in the above mentioned classic monographs, the authors made the following comments:

M. Van Dyke (1964, 1975):
“......, Lighthill (1949a) described a general technique for removing nonuniformities from perturbation solutions of nonlinear problems...... An analogous straining of the independent variable was used by Poincare (1892) to obtain periodic solutions of nonlinear ordinary differential equations. For this reason Tsien (1956), in a survey article, has dubbed it the “PLK method,”, the K standing for an application to viscous flows undertaken by Kuo (1953, 1956). We prefer to speak of Lighthill’s technique, or of the method of strained coordinates, which describes its essential feature.”

J. Kevorkian and J. D. Cole (1981):
“Method of Strained Coordinates (Lindstedt’s Method) for Periodic Solutions: In the form we will consider this method, it was discussed in 1892 in volume II of Poincare’s famous treatise on celestial mechanics, Reference 3.1.1. Although Poincare gave due credit for the original idea to an obscure reference by Lindstedt in 1882, subsequent authors have generally referred to this as the method of Poincare. Actually, the idea goes further back to Stokes, Reference 3.1.2, who in 1847 used essential the same method to calculate periodic solutions for a weakly nonlinear wave propagation problem (cf. Problem 3.1.1). Strictly speaking, one should therefore refer to Stokes’ method. This has not been the case and many authors have called it the PLK method (P for Poincare, L for Lighthill who introduced a more general version in 1949, and K for Kuo who applied to viscous flow problems in 1953). To minimize confusion, we will adhere to Van Dyke’s nomenclature of the ‘method of strained coordinates’ and refer the reader to Van Dyke (1975) which contains an extensive discussion of applications in fluid mechanics.”

Both comments were critical on associating Y. H. Kuo to the method not based on a new idea or an improved technique/method but based on “an application”. Such an inappropriate association was originated from the following paper:
Tsien, H. S., 1956: The Poincare-Lighthill-Kuo method. Adv. Appl. Mech., 4, 281-349.

I believe there is an important issue on the method of strained coordinates, or more general methods of singular perturbation, that is rarely clarified in its many applications.......

Van Dyke, M., 1975: Perturbation Methods in Fluid Mechanics. Annotated Edition. The Parabolic Press, Stanford, Calif., 271 pp.
Kevorkian, J., and J. D. Cole, 1981: Perturbation Methods in Applied Mathematics. Springer-Verlag, New York, 558 pp.

coolboy

One common trick used in the singular perturbation methods is to eliminate the so-called “secular” terms. By setting the coefficients of those resonantly forced terms to zero, the unknown constants in the “strained coordinates” can be determined. While doing so, one has already assumed that the solution of a given differential equation will be finite or the solution is of quasi-periodic nature. Assuming a solution u(t,[e]) being a function of both the independent variable t and a small parameter [epsilon]=[e], the solution being finite with respect to t and the convergence of its asymptotic expansion with respect to [e] are two different issues. Hence, it is necessary to clarify that the differential equation to be solved only contains finite or quasi-periodic solution in t within the interested parametric domain before applying the singular perturbation methods to solve it.

Sometimes, one may understand the qualitative behavior of the solution for a differential equation without actually solving it. The so-called geometric method via phase diagram has especially been well developed for the ordinary differential equations. Though mathematical handbooks generally list various methods to solve ordinary differential equations, many textbooks in ordinary differential equations often emphasize the geometric aspects of the ordinary differential equations. The following books on my bookshelves are such examples:
Arnol'd, V. I., 1973: Ordinary Differential Equations. MIT press, 280 pp.
Arrowsmith, D. K., and C. M. Place, 1990: An Introduction to Dynamical Systems. Cambridge Univ. Press, Cambridge, 423 pp.
Jordan, D. W., and P. Smith, 1977: Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford, 360 pp.

coolboy

It can often be argued that a finite or a quasi-periodic solution of a system implies the existence of an invariant or a conserved quantity (more general and exact statement is Noether's theorem). It is well known that the total energy of a physical system usually is a conserved quantity, i.e., it does not change as the system evolves with time. As a result, one particular conserved quantity in a physical system generally corresponds to the total energy. The total energy of a mechanical system is often called Hamiltonian. Furthermore, the so-called “Hamiltonian mechanics” can be considered to study mechanics, including both the particle mechanics and fluid/continuum mechanics, from a geometric perspective and in terms of their qualitative behaviors. In general, ordinary differential equations describe motions of particles whereas partial differential equations describe fluid mechanics. I consider the following books on my bookshelves, especially the last three, to be good reference books of using Hamiltonian mechanics to study the global and qualitative behavior of the fluid dynamics:
Abraham, R., J. E. Marsden, and T. Ratiu, 1988: Manifolds, Tensor Analysis, and Applications. Second edition. Springer-Verlag, New York, 654 pp.
Arnol'd, V. I., 1978: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 462 pp.
Arnol'd, V. I., 2014: Vladimir I. Arnold Collected Works. Volume II: Hydrodynamics, Bifurcation Theory and Algebraic Geometry (1965-1972). Springer-Verlag, Beilin, 464 pp.
Swater, G. E., 2000: Introduction to Hamiltonian Fluid Dynamics and Stability Theory. Chapman & Hall/CRC, New York, 274 pp.
黄思训、伍荣生编著，2001：《大气科学中的数学物理问题》，气象出版社，北京，540页。

coolboy

Let me tell a few interesting stories about nonlinear stability studies without complete references or sources and thus can be considered as anecdotes. I mentioned previously in this post (32#) that:
+++++++++++++
I first learnt the beauty or the power of variational method in the following book:
Morel, P. (Editor), 1973: Dynamic Meteorology. D. Reidel Publishing Company, Boston, 622 pp.
The book contained a set of lecture notes delivered by then six world renowned atmospheric dynamists. One of those was by J. G. Charney:
Charney, J. G., 1973: Planetary Fluid Dynamics. In: “Dynamic Meteorology,” P. Morel (Ed.), pp. 97-352. D. Reidel Publishing Company, Boston.
in which the author derived stability/instability criterions for a circular vortex by the variational method.
+++++++++++++

Charney's introduction of using variational method to study fluid stability in his lecture notes followed the seminal work by R. Fjortoft as shown in the following paper:
Fjortoft, R., 1950: Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofys. Publ. 17(6), 5-52.

In early 1950s, Charney and Fjortoft were close collaborators and they were major contributors to implementing numerical weather prediction right after the invention or availability of electronic computer. J. Charney and R. Fjortoft were considered the best scientists in the field of dynamic meteorology at the time. It was said that Fjortoft was considered smarter and more creative than Charney at the time. For example, in addition to his 1950 paper, his another paper on spectral distribution of kinetic energy remains a piece of significant work even in the modern time:
Fjortoft, R., 1953: On the changes in the spectral distribution of kinetic energy for two dimensional, nondivergent flow. Tellus, 5, 225-230.

Later, J. Charney continued his research career along the dynamic meteorology and became the best dynamic meteorologist in his time. On the other hand, R. Fjortoft launched a completely new research project in the mid-1950s. Based on a large funding resource he was able to acquire and also on his science vision at the time he invested his efforts into inventing a new type of computer, if I recall correctly, called optical computer. Later, his science project failed completely and his 1953 paper was probably the last science paper indicating that he was once a serious scientist. [创新有风险，改行多失败。"路线方向错了，则可丢失一切。"]

coolboy

Variational method generally leads to sufficient conditions for stability or necessary conditions for instability. This might be part of the reason that variational method was not as popular as the alternative eigenvalue analysis to investigate the linear instability in fluid mechanics for the latter generally leads to both the necessary and sufficient conditions for instability. Another interesting and significant progress after Fjortoft's 1950 paper was the one by V. I. Arnold:
Arnold, V. I., 1965: Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Doklady Mat. Nauk., 162, 773-777.

Arnold was already a famous mathematician at the time so his 1965 paper was immediately well received in the science community. Arnold's (1965) approach was mathematically more general and formal in terms of functional variation whereas Fjortoft approach appeared being rooted in the traditional stability analysis based on parcel displacement. More importantly, Arnold claimed that his method dealt with “nonlinear stability” or stability of flow “with respect to finite perturbations”.

After carefully reading and digesting Arnold's paper (1965), several serious researchers understood that Arnold's (1965) variational method only led to the same conclusion of a linear stability (with respect to infinitesimal perturbations) as the one by Fjortoft (1950). Some of the researchers tried (but was still considered to fail) to extend Arnold's conclusion into a nonlinear one by introducing “mean value theorem” in the derivations. In early 1980s, there was a well-known fluid mechanist, calling him “Professor X”, who thought he found a way to truly extend Arnold's linear stability criterion into a nonlinear one. While preparing his draft, he was told that Arnold derived nonlinear stability criterion already. He checked a few other papers by Arnold and still could not convince himself that Arnold solved the problem already. However, there was yet another short paper by Arnold that was not easy to find in most libraries:
Arnold, V. I., 1966: On an a priori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Uchebn. Zaved. Matematika, 54, no. 5, 3-5. (English Transl: Am. Math. Soc. Transl., 79 (1969), 267-269.)

Note that the above paper not only was short (3 pages) but also was translated into English three years after its publication in Russian. That was mostly because the journal was not well known to Western science community at the time. “Professor X” finally got this 3-page paper and discovered the most important contributions made by V. I. Arnold to the field of fluid mechanics. “Professor X” made significant contributions to the science community in popularizing Arnold's nonlinear stability theorems. The contents of Arnold's (1966) paper were also included in his 1978 textbook (Russian edition published in 1974):

Arnold, V. I., 1978: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 462 pp.