Physical limitations of mathematical constructions are discussed, which should be taken into account in developing or modifying mathematical models. We begin with consideration of the method of describing physical objects using numbers and restrictions followed from this method. Next, we formulate some general recommendations concerning procedures for modifying mathematical models. Since models of physical phenomena are considered, it is natural to provide a physical interpretation for each stage of the model development. Unfortunately, some of transformations used are treated as purely technical tricks, and therefore the question of the physical meaning is not raised in such cases. The lack of physical meaning of some mathematical procedures does not make them unambiguously unacceptable. However, this marks out the place that requires a reasonable interpretation because the final result should possess the physical meaning. Finally, we discuss the issues related to the dimensionality of the space of places of a model. The above-mentioned physical limitations often are left without necessary attention. Sometimes this leads to various undesirable consequences, which may include excessive complication of the problem, an implicit substitution of the declared problem with another one or, finally, the absence of solution of the formulated problem.
Mathematical models physical meaning dimensionality of the space of places
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The author is grateful to his colleagues Dr A Holmstock and Dr V Ryabchenko for helpful discussions and valuable comments. This research was carried out in the framework of the theme 0149-2019-0015 of the Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russia.
A N Kolmogorov and S V Fomin, Introductory real analysis (Dover Publications, New York, 1975)Google Scholar
Here ‘fractional’ means that the corresponding physical dimension of a density is a ratio of two measures of two different standards. It is clear that any such ratio may be given a special name (recall e.g. ampere, knot, etc.). However, this does not make such quantity a measure, i.e. an integral parameter. It still remains a differential parameter or a density.Google Scholar
V E Zakharov, Zh. Prikl. Mekh. Tekh. Fiz.9(2), 86 (1968); English: J. Appl. Mech. Tech. Phys.9(2), 190 (1968\(/\)1972)Google Scholar
H C Yuen and B M Lake, Advances in applied mechanics (Academic Press, New York, 1982) Vol. 22, pp. 67–229Google Scholar
S Bochner and K Chandrasekharan, Fourier transforms (Princeton University Press, Princeton, 1949)zbMATHGoogle Scholar
R D Richtmyer, Principles of advanced mathematical physics (Springer, New York, 1978) Vol. 1CrossRefGoogle Scholar
A G Kurosh, Lectures on general algebra (New York, 1963) p. 335Google Scholar
J O Hinze, Turbulence. An introduction to its mechanism and theory (McGraw-Hill, New York, 1959)Google Scholar
A S Monin and A M Yaglom, Statistical fluid mechanics (MIT Press, Cambridge, 1971) Vol. 1Google Scholar
The commonly used experimental determination (i.e. measurements) of various local parameters (velocity, mass density, temperature, pressure, etc.) should not be misleading. In each such case, we measure not the limit of ratio of measures at some point of continuum, but the ratio of measures of small but finite volume of real fluids. Since any such volume always contains infinitely many points of the medium, each measurement itself actually is the averaging over this volume.Google Scholar
C Truesdell, First course in rational continuum mechanics (Academic Press, Cambridge, 1977)zbMATHGoogle Scholar
J Serrin, Mathematical principles of classical fluid mechanics, in: Handbuch der physik, Bd VIII\(/\)1 (Springer, Berlin, 1959) pp. 125–263Google Scholar
Introduction of complex numbers may retain physical meaning if the real and imaginary parts possess their original physical meaning separately.Google Scholar
The task of physics, and of science in general, is the development of means for describing observational data in order to be able to predict this kind of data. The science, which deals with something that is not observed neither directly nor indirectly, does not exist by definition.Google Scholar
T G Elizarova and Yu V Sheretov, J. Comput. Math. Math. Phys.41, 219 (2001)Google Scholar
By differentiation, the calculation of the ratio of the increment of a function under a time step to the magnitude of this step is meant here.Google Scholar
The similar expression corresponds to the case of the Lagrangian coordinates.Google Scholar
Generally speaking, a vector space with uncountable dimension.Google Scholar
M Belevich, Proc. RSHU38, 59 (2015) (in Russian); Proc. RSHU40, 81 (2015) (in Russian)Google Scholar
B F Schutz, Geometrical methods of mathematical physics (Cambridge University Press, Cambridge, 1980) p. 250CrossRefGoogle Scholar