книги / Механика и прикладная математика логика и особенности приложений математики
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БЛЕХМАН Илья Израилевич, МЫШКИС Анатолий Дмитриевич, ПАНОВКО Яков Гилелевич
МЕХАНИКА И ПРИКЛАДНАЯ МАТЕМАТИКА
Логика и особенности приложений математики
Заведующий редакцией Л . А. Русаков. Редакторы А. Г. Мордвинцев, Д. С. Фурманов
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S Y N O P S I S
of the book «Mechanics and Applied Mathematics: The Logic and Characteristic Features of Mathematical Applications» by L I. Blekhman, A. D. Myshkis and Y. G. Panovko (2nd edition, Moscow, «Nauka», 1990, 350 pages)
1* INFORMATION ABOUT THE AUTHORS
L I. Blekhman (b. 1928)*—professor, Doctor of Physical and Mathemati cal Sciences, head of the fundamental studies department of the All-Union Re search and Development Institute of Mineral Resources (Leningrad). Member of the National Committee of the USSR for Theoretical and Applied Mechanics. Member of three scientific councils of the USSR Academy of Sciences. Collabo rator of the journal «Applied Mathematics and Mechanics»
One of the leading specialists in the domain of applied mathematics and mechanics, non-linear oscillations and vibration engineering, he has laid the foundations of vibration displacement theory and of revolving bodies synchroni zation theory, discovered and studied the phenomenon of self-synchronization of unbalanced rotors. I. I. Blekhman is the author of the following monographs: «Vibration Displacement» (in collaboration with G. JL Janelidze), «Dynamic Systems Synchronization», «Synchronization in Nature and Engineering», «What can Vibration do?» and also of more than 150 papers. He is editor of the second volume of the «Vibration in Engineering» handbook in 6 volumes. Several of his books and articles have been translated into foreign languages.
I. I. Blekhman is the author of a number of inventions. Some of the leading firms in the USA, Japan, Bulgaria and Iran have bought the right of using them.
A. D. Myshkis (b. 1920)—Doctor of Physical and Mathematical sciences, professor of the higher mathematics department of the Moscow Railway Trans port Institute.
His scientific interests lie basically in the domain of theory and application of differential equations and adjacent fields of mathematics. His published works include more than 230 papers, 10 monographs and textbooks translated into 10 languages. Among these—«Linear Differential Equations with a Delayed Argument» (3 editions), «Elements of Applied Mathematics» (in collaboration
with |
academician Y. |
B. Zeldovitch—8 editions), «Lectures in Higher Mathema |
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tics» |
(9 editions), «Hydrodynamics in Zero-Gravity State» |
(in collaboration — |
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2 editions). |
(b. 1913)—corresponding member of |
Latvian SSR Aca |
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Ya. G. Panovko |
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demy of Sciences, doctor of technical sciences, professor of the Theoretical mec hanics department of the Leningrad Shipbuilding Institute, member of the Na tional Committee of the USSR for Theoretical and Applied mechanics, member of two scientific councils of the USSR Academy of Sciences.
Y. G. Panovko is the author of numerous publications on stability and vibrations in elastic systems. His published monographs include: «Statics of Elastic Thin-Walled Rods» (in collaboration with G. J. Janelidze), «Basic Principles of the Applied Theory of Elastic Systems Vibrations» (5 editions), «Stability and Vibrations of Elastic Systems» (in collaboration with I. I. Guba nova—6 editions), «Introduction to the Mechanical Vibrations Theory» (3 edi tions), «Introduction into Mechanical Collision Theory», «Mechanics of the De formed Solid Bodies» as well as a number of papers in academic journals, some of which were published in English.
2. ABSTRACT
In this book the authors have endeavored to present a systematic review of the basic principles of applying mathematics for solving practical problems (especially in the domain of mechanics) and of the typical methods of reasoning in this process. Emphasis has been laid on the idea that the logic of applied stu dies based on mathematics is essentially different (and cannot be but different)
from classical formal logic, characteristic of «pure» mathematics. The foregi*wrfK in applied mathematics is taken not by the rigidly deductive method, but by «rational» reasoning (in the well-known book by G. Polya it is called «plausible reasoning»).
The specific character of applied mathematics is universally acknowled ged—many valuable ideas on this subject have been published in books, scientific papers and proceedings of numerous conferences; a number of interes ting and keen observations expressed in the course of discussions remain un published and form a store of oral «folklore». Up to now this rich fund of ideas has not been properly summed up, integrated into a coherent whole and trans formed into engineering principles. The authors have endeavored to achieve this task. In so doing they have drawn freely from the works of numerous predeces sors (more than 600 names of other scholars are mentioned in the text) and also from their own experience of research and teaching activity.
Although the majority of examples have been taken from the field of mecha nics, the basic treatment is of a general character representing in fact the basic philosophy of applied mathematical research in the widest meaning of the word. This is also why the style and the language of the book approach the language and style of humanities.
Much attention is devoted to the still unsettled issue concerning the degree of mathematical rigour necessary in applied research. Throughout the book it is stressed that any plausible motivation for the correctness of results and reaso ning may be accepted as a sufficient proof, whereas mathematical rigour in app lied research is not an end in itself, but only a way of avoiding gross mistakes.
The book provides a thorough discussion of the evergreen problem of crea ting an adequate mathematical model for solving an applied problem. The re quirements of simplicity and optimization 6f the model are treated as well as the role of a simplifying hypothesis (as, for instance that of linearity of some of the initial relations), the importance of standard schemes, the question of the model’s determinancy or stochasticity and its adaptability for computers. A spe cial attention is paid to some effective analytic procedures and a convincing in terpretation of the results achieved.
A separate chapter is almost wholly devoted to psychological problems in volved, i. e. the inertness of thinking, mistakes in choosing adequate models and research procedures, as well as to the problems concerning teaching mathe matics to specialists in mechanics or other branches of teclmology within the framework of the University curriculum.
The book was first published in 1983. Its 6000 copies were immediately sold out, it became a rarity. Specialists accepted it with interest. It received strong support on the part of the vice-president of the USSR Academy of Scien ces К. V. Frolov and academicians N. M. Moiseev and V. V. Novogilov. An extended and favourable review written by I. Grekova (pen-name of E. S. Ventzel, professor of Applied Mathematics and doctor of Technical Sciences) was pub lished in the magazine «Science and Life».
The present second edition has been substantially revised, although its main conception remained intact. The new edition is made up-to-date with some additional material and a number of new references.
3. C O N T E N T S |
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Forewor d |
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Introduction |
1. The logic of |
applied |
mathematics |
(105 pages) |
C h a p t e r |
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§ 1. Applied |
and theoretical |
trends in |
mathematical |
development (Two |
main sources of mathematics; applied and theoretical trends. The origins. Scientific Renaissance. The period of set theory domination. A review of the present-day situation. What shall mathematics include? Various points of view on applied mathematics).
§ 2. On the difference of some approaches in pure and applied mathematics. (Preliminary observations. «Existance» in pure and applied mathematics. The infinity problem. Applied mathematics and number. A note on impossible events. The rate of convergence in approximation methods. On the concept of function. Stability in relation to the change of parameters. Fuzzy concepts. On the application of substantial notions and reasoning. On different trends in
the solution process. On mathematical rigour. Examples. A few |
quotations. |
§ 3. Rational reasoning. (The concept of rational reasoning. |
Examples of |
rational reasoning and their peculiarities. Types of rational reasoning. Deduc tion elements of rational reasoning. Degree of accuracy and probability. Control and increase of plausibility. On practical fidelity. Rational reasoning from the
point of |
view |
of |
optimization). |
mathematical research when |
solving |
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C h a p t e r |
2. |
Stages of applied |
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problems |
of |
mechanics (130 pages). |
the problem. (Preliminary |
remarks. |
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§ 4. |
The |
mathematical statement of |
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On the concept of model in applied research. The adequacy require ment. The influence of omitted factors. The simplicity and optimization, requirements. Phenomenological and semi-empirical laws. The defining parameters and the number of the degrees of freedom. The hierarchy of variables. A direct seperation of movements in non-linear mechanics as an example of the hierarchy of vari ables determination. On the control of the model. Further comments on modell ing in ^mechanics). f
§5. Choice of the methods of research. (External and internal plausibility. Comments on the interaction of technologists and mathematicians. On the role of estimates. On choosing the degree of accuracy. Discreteness and continuum. The role of a linearity hypothesis. Determinancy and randomness. Stability. Introduction of a small parameter. Interpolation and extrapolation. Further comments on deduction. The importance of examples. Improving accuracy. Computers. Addenda. Volitional decisions).
§6. Analysis and interpretation of mathematical results. (Preliminary remarks. General corroboration of research. Search for the unexpected. Presen tation of results.)
C h a p t e r |
3. Some subjective problems (45 pages) |
§ 7. Mistakes. |
(Psychological barriers and inertia of reasoning Mistakes |
in the choice of models. Mistakes in the choice of method. Mathematical errors.) § 8. Problems of special training. (Mathematical education for engineers.
The development of mathematical intuition. Methods of reasoning. Search for acceptable solutions. On formal calculations and exercises. On the mathemati cal curriculum for engineers. On teaching mechanics. On the training of the specialists in applied mathematics. On publications).
Notes |
to chapters |
Bibliography |
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Index |
of authors |
Index |
of some concepts and terms |
The book contains 350pages