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I.Kant
Appendices
As appendices the author has purposely placed responses which have acted(arrived) on published clauses(articles), and also on a pre-print of the grant(manual). Also it is made at all because in responses support of sights of the author is expressed. Responses represent the big cognitive interest as from them it is distinctly visible, on what way there should be basic researches in the field of mechanics of a deformable firm body. Besides the experts who have signed responses, that not knowing, the statements have actually revived that long, but not finished discussion which conducted among themselves D'Alembert and Euler. It is easy to be convinced of it, having familiarized with a response of experts of institute of mechanics NASU (N.P.Plahtienko and J.J.Antonjuk) on clause(article) Paradoxes of classical decisions of the wave equation (the Appendix 6). In a response the basic reviewers assert point of DAlemberts view, and in general supported response other expert (J.J.Kajuk) nevertheless states the special position similar to a position of Euler. It is uneasy to notice, what even the official conclusion of a scientific seminar of institute of mechanics NASU (the Appendix 3) is incompatible with reviews of the mentioned above experts and specified among acted on a seminar (Appendices 2 and 6).
For this reason the author believes, that long-term discussion D'Alembert and Euler in which after a while have taken part Bernoulli D. and Lagrange, it is impossible to count finished. The publication of responses the author also pursues the purpose to draw attention of successors of our oustanding predecessors to that him(it) honour to continue this discussion and, probably has dropped out, to define(determine), where all the same there is a true. Without continuation of such discussion to which necessity some reviewers pay attention also, revision of the doubtful heritage which have collected during centuries as mathematical misunderstanding and physical paradoxes in decisions many(much) classical and, probably, still a lot of nonclassical problems(tasks), appears rather problematic.
It would be desirable, that experts at whom remarks on the problems mentioned in the grant(manual) or offers on rendering assistance to the author in acceleration of the edition of the subsequent parts PARADOXES of MCM will appear, have sent their author by e-mail continuum-paradoxes@yandex.ru or have phoned (044) 526-42-57, Kiev. The review can direct the Ministry of Education and sciences of Ukraine (a copy in electronic variant to the author) with the purpose of their publication in copies of the basic circulation or in the subsequent editions of the manual. For prevention of distortions of reviews in English-speaking variant of the manual it is expedient to apply(put) their translation into the English language. The publication of reviews, the author is sure in it, it will be possible to revive well-known discussion of our oustanding predecessors.
The appendix 1
Response
about A.A.Kozachok's article Paradoxical features of the fundamental classical equations of dynamics(changes) of deformable environments , published in the collection of scientific works the Bulletin of National technical university the Kharkov polytechnical institute , 2001, release 129, page 272-283
In article the following is argued enough. The standard assumptions of small size of an error of the fundamental equations of the classical theory of elasticity are allowable only in some cases. For such cases number of the Move (the attitude(relation) of speed of fluctuation to speed of a sound) and deformation - sizes of one order. These features are noticed on simple examples in an one-dimensional case. Then the author has distributed these features to three-dimensional parities(ratio). For this purpose it(he) has presented the common decision by known transformations of the three-dimensional wave equations.
mentioned actually means inevitability of revision of numerous results in various areas of mechanics of a deformable firm body. It especially concerns those problems(tasks) when the number of the Move can appear considerable: explosive loadings, powerful ultrasonic fields, fluctuations near to resonances, etc.
Realization of such revision is connected to necessity of realization of the following actions:
Attraction of experts of the National Academy of sciences of Ukraine and corresponding faculties of the higher school to a urgency of carrying out of researches for delimitation of applicability of known classical decisions of problems(tasks) of the theory of elasticity; first of all it concerns problems(tasks) which were included in manuals;
Carrying out of basic researches on development of methods of the decision of the equations of the theory of elasticity at preservation of nonlinear members at components of acceleration;
The organization of discussions on the given problem on pages of one of conducting periodicals;
Preparation of inquiries about budgetary and sponsor's assignments; it is necessary to involve the international funds, for formation of mobile scientists on the marked directions of researches.
The conducting scientific employee
Institute of a hydromechanics National
academies of sciences of Ukraine
the doctor of physical and mathematical sciences,
the professor V.V.Meleshko
To the academician - secretary of Branch of mechanics to academician Pilipenko V.V.
Dear Victor Vasiljevich! To me Alexander Antonovich Kozachok has addressed. He has paid my attention to discrepancy of mechanics of a firm body and hydrodynamics. I in this question of him completely support. Now he has remarks concerning fluctuations of a string. I ask to pay him attention.
The academician of the National academy of sciences of Ukraine V.G. Barjahtar
-------------------------------------------------------------------------------------
I agree with a response of professor V.V.Meleshko.
The conducting scientific employee of Institute
mechanics of the National academy of sciences of Ukraine,
the doctor of physical and mathematical sciences N.P.Plahtienko
21.09.01
-------------------------------------------------------------------------------------
Offers of professor V.V.Meleshko are worthy also them it is necessary to support comprehensively.
Managing faculty of theoretical mechanics
National technical university
Ukraine the Kiev polytechnical institute ,
dr.Sci.Tech., the professor Pavlovsky M.A.
05.11.01.
-------------------------------------------------------------------------------------
I agree with V.V.Meleshko's response.
The leading expert of institute of mathematics
National academy of sciences of Ukraine,
the doctor of physical and mathematical sciences,
professor Trotsenko V.A.
11.12.01.
-------------------------------------------------------------------------------------
The analysis of bases of mechanics of continuous environments is of interest and should be supported.
The head of a department of institute of a hydromechanics
National academy of sciences of Ukraine
the doctor of physical and mathematical sciences,
professor Selezov I.T.
12.03.2002
-------------------------------------------------------------------------------------
Offered by Kozachok A.A .improvements of the equations of the dynamic theory of elasticity are worthy.
The academician of an academy of sciences of the higher school,
professor Goroshko O.A.
1.10.02.
The appendix 2
Conclusions
to A.A.Kozachok's article Paradoxical features of the fundamental classical equations of dynamics of deformable environments
As well all know, for studying natural phenomena build mathematical models. Models prove on the basis of the intuitive reasonings, the known data from the physical processes, the given experiments, practice, etc. For an estimation of reliability of model from mathematical positions, borders of its(her) application use widely used approach. Build more the general(common), nonlinear model. In comparison with this model establish, that the linear model gives. What borders define(determine) its(her) applications. Thus for quantitative and quality standard widely use characteristic parameters. Such parameters are most peculiar for the given processes and the phenomena. For example, processes of distribution of all sort of signals (thermal, electromagnetic, gravitational, etc.) are frequently considered(examined) In a space (anyway, within the limits of Solar system). In such cases speed of light extremely important characteristic. It is possible to consider(examine) processes in the gas environment which surrounds the Earth. Swift-flowing processes always need to be investigated in comparison of speed of their realization with speed of distribution of small indignations in a considered(an examined) point of the gas environment. We already well know, how oscillatory (wave) processes proceed at speeds of smaller and big speeds of a sound. These are different things!
And now we shall pass to the firm deformed body. Well-known, that in a firm body different laws of distribution of waves are investigated. Different types of these waves are investigated. It too environment(Wednesday) and is completely fair in it(her) (be relative to it(her)) to put such questions. She(it) can pass(miss) what waves? What is necessary for this purpose mathematical model? What will take place at speed of distribution less or more speeds of a sound? Or will take place nothing, any phenomena will not arise?
Also that we have. Long years for the description of these waves (their different types!!) Use the linear equations. But they cannot give the answer to the put question. Business was eclipsed with what situation. The following has already become a habit to believe. By consideration of small deformations of indignation in the environment Euler's approaches and 03@0=60 almost coincide. And, probably, for the first time senior lecturer A.A.Kozachok in article Paradoxical features of the fundamental classical equations of dynamics(changes) of deformable environments has paid attention and "has turned" our traditional thinking. It(he) asserts(approves) the following. At desire to write the certain equations for the description of wave processes in the deformed firm body it is necessary to formulate all over again all in Euler's variables (as well as in a liquid, gases). Then it is necessary to consider(examine) different simplifications in view of number of the Move for the specified environment! Appeared, that here the important role is played with record of acceleration (absolute acceleration) in Euler's variables. This record is expressed through the certain derivatives on time from a vector of moving. The vector of moving also should be expressed through variables - Euler's coordinates! The author of these lines has become interested in such question. And somebody from known coryphaeuses (Sedov, Ilyushin, Goldenblat, Rahmatullin, Noll, Truesdell and it it is a lot of others) calculated obvious expression for the specified acceleration generally? Appeared, that nobody calculated. Why? Because this expression bulky enough. Calculation of this acceleration laborious, long also demands attention. The author of a response has calculated this acceleration. As a special case (for an one-dimensional case) the author of a response has received that has in another way made A.A.Kozachok. The author of a response has confirmed with it his(its) researches. And the most basic necessity of carrying out of such researches!
So appeared? We, and before A.A.Kozachok had been calculated expression for acceleration material G0AB8G5:. Acceleration is calculated through the certain derivatives on time from moving. Moving too depends on Euler's variables. The following has been established. Expression for acceleration will be entered with numbers of the Move and other parameters for Wednesday of the deformed body. On the basis of various assumptions it is possible to receive different linear and nonlinear variants of the wave equations (for distribution of waves of various intensity).
Finishing(Stopping), once again I shall note the following. The modest researches senior lecturer A.A.Kozachok 1C48B us from the settled approaches of " long-term dream . He has shown the following. At the analysis of wave processes in the deformed body it is necessary to formulate all over again all (as in a liquid, gas dynamics) in Euler's variables. Then it is possible to build variants of models for different numbers of the Move and other parameters!
Certainly, A.A.Kozachok not the graduate of university . Therefore his aspiration to give mathematical estimations causes criticism. And basically, his approach to estimations - correct. It is possible to approve researches of senior lecturer A.A.Kozachok in the specified direction.
The professor, the doctor of physical and mathematical sciences,
acting as the head of a department of Institute of mechanics by S.P. Timoshenko of the National academy of sciences of Ukraine J.F. Kajuk. 10.10.01
The appendix 3
Extract
from the report of session of a seminar on mechanics of Institute of mechanics
by S.P.Timoshenko of the National academy of sciences of Ukraine
On October, 31, 2002 !3
Kiev
Are present:
academicians of the National academy of sciences of Ukraine: A.N.Guz ., J.N.Shevchenko;
members - correspondents of the National academy of sciences of Ukraine: V.D.Kubenko,
L.P.Horoshun, N.A.Shulga;
doctors of sciences: V.A.Zarutsky, J.N.Podilchuk, Century. G.Karnauhov, Century. A.Maksimjuk, . P.Plahtienko, J.F.Kajuk, V.M.Chehov, A.P.Zhuk, I.S.Chernyshenko, V.G.Savchenko, And. Babayev, B.P.Maslov, J.J.Rushchitsky, K.I.Shnerenko, V.I.Kozlov, I.J.Babich, V.P.Golub, A.A.Kaminsky, V.B.Larin, I.F.Kirichok, A.T.Vasilenko;
candidates of sciences.
Listened:
A.A.Kozachok's report on a theme: Paradoxical features of the fundamental classical equations of dynamics of deformable environments and offers of experts concerning their doubtful decisions .
Questions have set:
A.N.Guz, V.D.Kubenko, J.N.Shevchenko, L.P.Horoshun, J.J.Rushchitsky, A.P.Zhuk
Have acted:
J.F.Kajuk, N.P.Plahtienko, A.N.Guz.
Have decided:
1. In the message acting has stated the interpretation of results of scientific researches of other scientists.
2. Essentially new results it is not revealed.
3. Probably, results of the lecturer have methodical value. Under the statement of the Lecturer his(its) point of view some scientists of the Kiev national university of name Taras Shevchenko and National technical university Kiev polytechnical institute are support .
4. In connection with above mentioned to recommend him to give the report at methodical seminars of the specified universities.
5. At sessions of a seminar of institute of mechanics to this question further to not come back.
Chairman of session,
the academician of National
academies of sciences of Ukraine A.N.Guz
The scientific secretary,
the doctor of physical and mathematical sciences A.P.Zhuk
The appendix 4
Response
for A.A.Kozachok's work Paradoxes of classical decisions of the wave equation , published in Bulletin of NTUU KPI "Mechanical engineering", v. 2, release. 38 2000
The author of article completely fairly specifies the following fact. In the textbooks quoted by him(it) under the theory of fluctuations and the equations of mathematical physics for the maximum(supreme) technical educational institutions at the formulation of a problem(task) about free longitudinal fluctuations homogeneous ?@87<0B8G5A:>3> a core with one fixed end incompatibility of initial and boundary conditions is supposed. Incompatibility takes place in a point of crossing of their ranges of definition. It is possible to give reason, however, for the following. For correct definition of a corresponding problem(task) it is enough to demand, that the decision satisfied to initial and boundary conditions everywhere, except for the specified point. As a matter of fact it also is done(made) by authors of the quoted textbooks without additional explanations.
Certainly, such statement of a problem(task) means the following. Corresponding decisions are not obliged to be any more smooth (or in other terminology classical). This fact is shown also by the author in the first part of clause(article). It(he) specifies on A8=3C;O@=>AB8 corresponding decisions.
It is necessary to note, nevertheless, the following. Such decisions (usually named generalized) can be approximated with any accuracy smooth decisions. Smooth decisions answer in appropriate way picked up coordinated initial and boundary conditions. The example of one such decision also is considered(examined) by the author in the second part of article.
As a whole, the considered work could be of interest for teachers and students of the maximum(supreme) technical educational institutions. However in work there is a following mere allegation of the author. The similar generalized decisions are "not physical", but they had already time enter under this name more modern textbooks on the equations of mathematical physics (in the considered case these decisions quite adequately reflect initial physical assumptions of an opportunity of the appendix of dot loading, and also of an opportunity of its(her) instant removal).
To lack of work it is necessary to attribute(relate) as well not adequate(answering) to a modern level of teaching of mathematics discussion of a fact of common knowledge. This fact consists in the following. Unreasonable rearrangement of operations of differentiation and a capture of the infinite sum can result in different results. This information is born in the beginning of clause(article). She(it) distracts attention of the reader.
The scientific employee of Institute of mathematics
National academy of sciences of Ukraine,
the candidate of physical and mathematical sciences G.V.Shchepanjuk
The appendix 5
I agree.
Deccan of physical and mathematical faculty
of National technical university
The Kiev polytechnical institute
the academician Barjahtar V.G.
Conclusions
concerning A.A. Kozachok's article Paradoxes of classical decisions of the wave equation, published in the periodic collection Bulletin NTUU KPI, Mechanical engineering , 2000, release. 38, B.2, pages124-133.
In article some exact decisions of the one-dimensional homogeneous wave equation as infinite numbers(lines) are analyzed. Such type of a problem(task) meet in many subject matters.
Defects of these decisions are noticed by the author, namely: divergences of the second derivative numbers(lines) and absence of such divergences at their preliminary summation, inconsistency of initial and boundary conditions and others really take place. Some positions cause doubts concerning conformity of the specified decisions to real processes.
At the same time some positions put forward by the author debatable or are insufficiently argued and specification demand.
In connection with stated the expediency of the following actions above follows:
Revision of other typical problems(tasks) of such plan which were included in textbooks;
Attraction of attention of experts and is especial teachers of the higher school to necessity of corresponding researches; it is necessary for correction of the noticed defects in decisions of educational problems(tasks);
Attraction of attention of students and post-graduate students at teaching different disciplines to erroneous positions in the specified decisions.
The professor of faculty of mathematics
National technical university
The Kiev polytechnical institute N.A.Virchenko
25.03.02
I agree with conclusions of professor Virchenko ...
Managing faculty of mathematical physics
The Kiev National university
name Taras Shevchenko, the doctor of physical and mathematical sciences,
the professor Samojlenko V.G.
26.04.01
With the resulted conclusions in A.A.Kozachok's article I agree, though, in my opinion, these cautions concerning bad convergence of numbers(lines) of derivatives already for a long time were marked in works of academician A.M.Krylov (1905-Math. Annalen, 1911-lectures on the approached calculations ). The merit of the author will consist in attraction of attention to these "thin" questions in educational problems(tasks).
The doctor of physical and mathematical sciences, the professor,
the conducting scientific employee of institute of a hydromechanics
National academy of sciences of Ukraine V.V.Meleshko
30.05.2001
I agree with conclusions of professor Virchenko N.A.if a problem(task) is put in classical sense it is necessary to specify conditions of the coordination of the initial and boundary data precisely. Such inconsistency results in introduction of concept of the generalized function and consideration of such problems(tasks) from positions of the generalized functions.
The doctor of physical and mathematical sciences, the professor, corresponding member
national academy of sciences of Ukraine, the head of a department
the differential equations in private(individual) derivatives
institute of mathematics of National academy
sciences of Ukraine Gorbachuk M.L.
8.06.01.
The appendix 6
Response
on A.A.Kozachok's article Paradoxes of classical decisions of the wave equation , published in the collection Bulletin NTUU KPI. Mechanical engineering , release. 38, v.2, 2000, pages 124-133
In article known exact decisions of the one-dimensional wave equation are analyzed. These decisions are used as typical educational problems(tasks) in mathematical physics, the theory of fluctuations, etc.
The author finds out such features of decisions - sawtooth character of movings of the end of a core, divergence of numbers(lines) for the second derivatives and others. Such features really take place and contradict the standard representations about a picture of mechanical fluctuations of elastic ph.
The noticed features of classical decisions of typical problems(tasks) can be shown and in other cases. If, for example, factors of members of trigonometrical lines to which the decision is submitted, contain squares of serial numbers of members EMBED Equation.3in a denominator. As an example a number(line) of others can serve and well investigated problem(task) about cross fluctuations of a string, and. These features are caused by incorrect statement of problems(tasks). At statement the basic classical requirement is not observed: a continuity and differentiability of the functions chosen for the description of initial and boundary conditions. A result such, incorrect, statements also appear bad, that is not physical decisions. These decisions also are exact from the mathematical point of view. However, these decisions are unsuitable for practical application.
The bad decisions received as a result of incorrect statement, generally speaking, can be corrected by smoothing (coordination) of initial and boundary conditions. However, such procedure represents independent and enough challenge. The author pays attention to this problem also.
In view of stated the following obvious conclusion follows. It is necessary to reconsider other problems(tasks) of such type, to develop more correct approaches to their statements. It will allow to receive suitable for practical use and consistent decisions from the physical point of view.
The leading expert of a department of dynamics of complex systems of Institute of mechanics
S.P.Timoshenko's name of the National academy of sciences of Ukraine
the doctor of physical and mathematical sciences N.P.Plahtienko
The senior employee of a department of dynamics of complex systems of Institute of mechanics
S.P.Timoshenko's name of the National academy of sciences of Ukraine
cand.Tech.Sci. E.J.Antonjuk
I agree basically with conclusions of the doctor of physical and mathematical sciences of N.P.Plahtienko and Cand.Tech.Sci. E.J.Antonjuk. In my opinion, cannot exist correlation between initial and boundary conditions. These are different things with various dimensions.
The professor, the doctor of physical and mathematical sciences,
acting as the head of a department
institute of mechanics of National academy
sciences of Ukraine by S.P.Timoshenko Kajuk J.F.
10.10.01.
I think expedient introduction of positions of researches of the author in subject matters.
Managing faculty of theoretical mechanics
National technical university of Ukraine
The Kiev polytechnics institute ,
the doctor of technical sciences, professor Pavlovsky M.A.
05.11.01.
I think expedient at reading a rate of the theory of fluctuations to pay attention students to possible(probable) "paradoxes" considered by A.A.Kozachok in case of use of technical theories outside area of their application.
The head of a department of the theory of fluctuations
and vibration reliability of institute of problems
durability of the National academy of sciences of Ukraine,
corresponding member of National academy
sciences of Ukraine,
the doctor of physical and mathematical sciences, professor Matveev V.V.
15.05.02.
The appendix 7
THE REVIEW
on A.A.Kozachok's manual " Paradoxes of mechanics continuous mediums.
Part 1. Questions of nonlinear dynamics continuous mediums
A.A.Kozachok's manual Paradoxes of mechanics of continuous mediums. A part 1. Questions of nonlinear dynamics(changes) of continuous environments it has been investigated by the faculty of faculty of theoretical and applied mechanics of mechanical and mathematical faculty of the Byelorussian state university and it is discussed on faculty meeting.
Problems touched in the manual concern to questions of a correctness of statement of regional boundary problems(tasks) of nonlinear dynamics(changes) of continuous environments,
"paradoxes" in our opinion considered by A.A.Kozachok matter at a conclusion of systems of the allowing(resolving) equations of the applied (technical) theories based on the decision of dynamic problems(tasks) of mechanics of continuous environments. Especially it concerns statement boundary and entry conditions. One of the main conclusions is the following. It is necessary to approach(suit) rather seriously and responsibly(crucially) to statement and mathematical formalization of applied problems(tasks).
The material of the manual can be used at reading special rates. Such rates concern the general(common) and private(individual) questions and features of mathematical modelling and mathematical formalization of applied dynamic problems(tasks) of !. In particular there is a theory of fluctuations and theories of elasticity. Conclusions of the manual will be interesting also at reading rates on numerical methods of the decision of the differential equations in private(individual) derivatives. For example, they are of interest for demonstration of importance of strict statement of regional problems(tasks) and close(attentive) use of a method of numbers(lines) at transition from differential statement of problems(tasks) to the formulation in the final form.
The questions stated in the given manual, in that or other measure are considered(examined) at reading various rates at our faculty (for example, the basic rate Mechanics of continuous environments , special courses the Theory of fluctuations , Problems(Tasks) of the dynamic theory of elasticity , Features of mathematical modeling in MDSB and others.). They to us were well-known and earlier. Therefore in use of the manual of A.A.Kozachok in educational process at mechanical and mathematical faculty we yet do not feel need. Interest can represent only full edition of materials. Speech about them goes in the conclusion and conclusions of the manual. In this case use of materials in educational and research process of students can be rather useful.
Deccan mechanical and mathematical faculty,
of the Byelorussian state university, the professor, the doctor of physical and mathematical sciences N.I.Jurchuk
Managing faculty of theoretical and applied mechanics,
the professor, the doctor of physical and mathematical sciences M.A.Zhuravkov
The professor of faculty of theoretical and applied mechanics, the doctor of physical and mathematical sciences M.D.Martynenko
The appendix 8
THE REVIEW
on A.A.Kozachok's scientific edition " Paradoxes of mechanics of continuous mediums "
In work the Author has stated the vision of well-known assumptions. Such assumptions are put in a basis of linear mechanics of a firm deformable body. In particular, the Author puts under doubt a way linearizations the equations of movement of a firm deformable body, according to which EMBED Equation.3. This approached equality is comprehensible at small deformations. At such assumption there is an exact decision of classical problem(task) Lamb (action of the concentrated force on border elastic half-spaces). From this decision in 0A8<?B>B8:5 it is possible to receive the decision of classical problem(task) of Boussinesq. It is obvious, that the Author could strengthen the statements. For this purpose it was necessary to make an estimation of size of nonlinear members which have neglected in problem(task) Lamb.
With the Author it is possible to agree in the following. Representation of decisions of dynamic problems(tasks) of mechanics of a firm deformable body as Fouriers numbers(lines) a method of division of variables really are incorrect. However, if to a problem(task) (1.6.1) (its(her) decision as lines Fourier the Author results) to apply integrated transformation Laplace on a time variable all disagreements of some Fourier will be removed(will be taken off). In this case the decision of a problem(task) is represented as lines on the reflected waves. [G.Abramovits and I.Stigan. The directory on special functions. The formula 29.3.69].
Let's notice also the following. In representation the Llama (1.5.4) vectors of elastic moving should apply an additional condition to unambiguity EMBED Equation.3. Therefore the decisions constructed by the author (1.5.7) are necessary for reconsidering in view of this condition.
Taking into account told is higher, A.A.Kozachok's work has debatable character.
The candidate of physical and mathematical sciences, the senior lecturer of faculty of mechanics
The Lvov national university
Ivan Franko's name V.A.Galazjuk
The signature of senior lecturer Galazjuk V.A. I certify:
The pro-rector on scientific work of the Lvov national university of a name of Ivan Franko the Doctor of Chemistry, the professor B.J.Kotur
The appendix 9
THE REVIEW
on a pre-print Kozachok Alexander Antonovich
Paradoxes of mechanics of continuous mediums".
Part 1: " Questions of nonlinear dynamics of continuous mediums"
A.A.Kozachok 's work " Paradoxes of mechanics of continuous mediums" (the Part 1) is devoted to consideration of questions of a correctness of statements of dynamic problems(tasks) of mechanics of the continuous environment, from the point of view of application Eilers and Lagranges approaches. In the historical plan, these problems were $.9
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^:`a$gddgdinvestigated by many known scientists within two centuries. They basically were displayed in original scientific publications. However in educational and methodical grants(manuals) results of these researches are covered insufficiently. From this point of view, it is hard to recognize scientific novelty and a urgency of work of the author. However, from the point of view of a technique of teaching of mechanics of the continuous environment in the Higher school, work of the author and the problems of borders of applicability of Lagranges and Eilers approaches lifted in it(her) and are worthy. In particular, it is necessary to give due to the author in the following. It(he) has managed to concentrate attention on "paradoxes" of known decisions. These "paradoxes" in opinion of the author are caused by approach 03@0=60 and inconsistency of initial and boundary conditions of a problem(task). It, for example, concerns a problem(task) about longitudinal fluctuation of a core with one fixed end. At the same time, in the given work it is a unique concrete applied problem(task). Besides the criticism the author of known decisions has no scientific value without the offer of own ways of elimination unreasonableness.
In end it would be possible to recommend to the author to state the scientific reasons in the scientific article. This clause(article) should be sent in one of the conducting foreign magazines devoted to mechanics of the continuousmediums.
Professor Vloh R.O. (institute of physical optics) The appendix 10
Response
on A.A.Kozachok's manual
Paradoxes of mechanics of continuous environments
The offered(suggested) manual, as a matter of fact, is devoted to the analysis of character of received mathematical models depending on the chosen system of coordinates (Euler and Lagrange). From these models as consequence(investigation), "incorrect" decisions turn out when character of deformation actually differs from assumed(prospective) at a choice of system of coordinates. We shall not stop in detail on the trivial results resulted enough. It is necessary to note, that "paradoxes" observable by the author are connected to attempt to use the limited parcels(sendings) and reasonings. Borders of applicability of these parcels(sendings) should be established in each case in sense of adequacy to the described phenomena.
It is not surprising, that the decision of the wave equation cannot provide the set initial deformation and moving simultaneously. For performance of these conditions it is simultaneously necessary to involve more complex(difficult) model. This model should take into account "boundary" effects.
Similarly business and with the decision near to a strict resonance is. These decisions can be constructed, for example, with application asymptotic methods, instead of cannot be constructed at all. Thus consideration of the models adequate(answering) to physical features of proceeding processes (see Bogoljubov N.N., Mitropolskij J.A.'s works) is necessary. Obviously, also, that at presence of essential nonlinearity the principle of superposition is not applicable.
We pass to the general(common) assessment of works. It is necessary to note the following. It is necessary to state the questions mentioned in work to students. It is necessary to state them in completely other treatment. It is necessary to discuss borders of applicability of this or that classical model. It is necessary to do(make) it similarly J.G.Panovko's to works. References to these works are absent. In offered(suggested) edition the grant(manual) cannot be useful for understanding and development of representations about models and methods of mechanics of the continuous environment as observable "paradoxes" are only ascertaining of the fact about ignoring borders of applicability of this or that model.
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