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2013 | OriginalPaper | Chapter

5. Analysis of Physical Variables

Author : Enzo Tonti

Published in: The Mathematical Structure of Classical and Relativistic Physics

Publisher: Springer New York

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Abstract

Physical variables are the foundation of the mathematical formulation of physics because they arise from the existence of quantitative attributes of physical systems. To analyse the mathematical structure of physical theories, it is convenient to start with a detailed analysis of their physical variables. In this chapter we will present two new classifications: the distinction between global variables and their rates and the distinction between configuration, source and energy variables. It is shown that global variables are associated with space elements, and the corresponding densities inherit this association. This analysis was pursued on approximately 180 variables collected in the appendix c.

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previous chapter Cell Complexes

next chapter Analysis of Physical Equations

Science and Hypothesis, p. 44.

For the definition of global variables see p. 106.

The term description is equivalent to the terms viewpoint and approach.

A clear distinction between the material and spatial descriptions can be found in Shames [209, p. 132], where the term field is substituted by the term control volume. The two descriptions are also clearly analysed by Hughes [97, p. 2]. In fluid dynamics the material description is also called a system (Shames [209, p. 78]) or referential (Chadwick, [39, p. 54]) or Lagrangian description (Hunter, [98, p. 23]). The field description is also called an Eulerian description (Hunter, [98, p. 27]) or spatial description (Chadwick, [39, p. 54], Eringen and Suhubi [64, p. 11]).

We will use this notation in Chap. 13.

In his engaging book Hunter wrote: Particles are mythical mathematical entities which in no sense are to be confused with atoms or molecules [98, p. 23].

Shames [209, p. 78].

Temple [223, p. 11].

Shames [209, p. 133].

de Groot and Mazur [49, p. 3].

See p. 493 for the meanings of symbols.

In mathematics, the distinction between constants, parameters and variables was introduced by Leibniz; see Bourbaki [22, p. 244].

All the physical variables considered in this book are collected in the List of Physical Variables at the end of the book; see p. 493.

Tonti [227, 228].

Penfield and Haus [175, p. 155].

Hallen [87, p. 1].

With the exception of reversible thermodynamics because it is the science of energy.

Lebesgue [131, p. 150].

“Physical variables which are directly measurable always appear as domain functions; …this may be a line domain (i.e. an interval), a plane domain or a domain of more than three dimensions …” Lebesgue [130, p. 20].

See p. 113.

Tolman [225, p. 239].

The French literature speaks about densité volumique, surfacique, linéique.

Lebesgue [131, p. 130].

Many physicists believe that quantities which are not measurable should not be used in physics. This statement is wrong, as expressed by many illustrious physicists. See Appendix C.

MacFarlane [146, p. 17].

The direct algebraic formulation of physical laws is the starting point of the cell method: see Tonti [230‐232, 234] and the papers quoted on the Web site discretephysics.dicar.units.it.

Landau and Lifshitz [125, Sect. 81].

Klein [114, p. 9].

See Chap. 8.

Bashmakova and Smimova [9].

We use the star on the letter W because the line integral in a force field is the virtual work.

This relation is an immediate consequence of the principle of action and reaction: see Milne-Thomson [160, p. 630], Chadwick [39, p. 85]. In contrast, other authors deduce this property from the equation of motion: see Billington and Tate [15, p. 67], Jaunzemis [106, p. 209].

For mass and particle number, see p. 118; for the impulse of a force see p. 243.

The fact that velocity line integral is associated with time instants while velocity is associated with time intervals is a consequence of a peculiar ambiguity of velocity in fluid dynamics, as we will explain in Sect. 12.2, p. 356.

See p. 389.

See the analysis of the displacement, p. 331.

See p. 356.

Redlich [191, p. 588].

See p. 35.

See Chap. 9 for a more detailed presentation.

The global variables and equations of particle dynamics are displayed in the diagram in Table 9.1 at p. 242.

As will be explained in Chap. 12, at p. 356.

See p. 387.

Since the term flux is used with many different meanings, see p. 30 for the restricted meaning we apply to this term.

This is the indefinite time integral of the Lagrangian function along a path: see Lur’é, [144, vol. II, p.705], Landau and Lifshitz [123, p. 138].

Lewis and Randal [137].

Redlich [191].

See Appendix B.

See for example Fleury and Mathieu [71, vol. 6], Brillouin [30], Fournet [73], Jouguet [110].

This symbol is used by Corson [43, p. 8].

See p. 287.

Jouguet [110, vol. II; p. 31].

Fournet [73, p. 49].

See the quotation of Maxwell on p. 12.

See Chap. 9, p. 255.

See p. 429.

See Chap. 15.

Tonti [228].

Fung [77, pp. 285, 347].

Möller [163, p. 6], Rosser [194, p. 17], Jackson [102, Sect. 11.9].

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Title: Analysis of Physical Variables
Author: Enzo Tonti
Publisher: Springer New York
Book: The Mathematical Structure of Classical and Relativistic Physics
Print ISBN: 978-1-4614-7421-0

Electronic ISBN: 978-1-4614-7422-7

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-1-4614-7422-7_5

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