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Energodynamic system of physical quantities and concepts


To not mix with SI, unifying UNITS (explanation).

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Kogan J.Sh.

What is the dimension of the physical quantity?

1. The dimension is a more objective concept than the unit
2. Definition of dimension of the physical quantity and its designation
3. Аdvantage of application of dimensions of physical quantities
4. Dimensional analysis as a method of validation of laws
5. Dimensional formula for the generalized coordinates of state
6. Lack of definition of dimension

The concept of "the dimension of the physical quantity" in the process of systematization of physical quantities is very important. In practice, it is often confused with the notion of "unit". It should be noted that in mathematics there is another, albeit similar-sounding, but has a different meaning concept called "number of measurements".

1. The dimension is a more objective concept than the unit

Units are well known, but pay attention to the practice of their application One can measure the speed in m/s, and it is possible - in km/h. The volume of water in the piping is measured in m3, the volume of water in the ocean in km3, the volume of the beverage in the bottle in liters, the amount of drug in the pipette in milliliters, and the volume of sellable oil in barrels. In metrological reference of A.Chertov (1990), for example, are 18 units for the volume, 20 units for the mass, 16 units for the pressure. The set of directories devoted to how to translate some of the other units to the same physical quantities. This disparity is explained by historical and ethnographic reasons, not to mention the considerations of elementary convenience to users.

Nature to all units is irrelevant, they thought up mens on the planet Earth in order to communicate with each other and understand each other. On any other planet inhabited by intelligent beings, if finally found, well familiar to us physical quantities will be used entirely different units of measurement. Remain the same except that the fundamental constants, such as the ratio of the circumference to its radius, although they will be named and designated otherwise.

It follows that the laws of science, that is, the coupling equations between physical quantities, must be analyzed not using units of measurement, which can be a lot different for the same physical quantity, but with the help of some other concepts single-valued for the same physical quantity. Such concepts and began to call dimensions.

However, after the introduction of this concept in the physics arised a lively discussion about what is considered the primary: dimension or unit. The essence of this discussion and conclusions are given in separate article.

2. Definition of dimension of the physical quantity and its designation

To quote the definition of the International Vocabulary of Metrology JCGM 200:2012: quantity dimension is "expression of the dependence of a quantity on the base quantities of a system of quantities as a product of powers of factors corresponding to the base quantities, omitting any numerical factor".

Base quantities in the International System of Quantities ISQ are length (symbol L), mass (symbol M), time (symbol T) electric current (symbol I), thermodynamic temperature (symbol Θ), the amount of substance (symbol N), luminous intensity (symbol J). Clarify the to the concepts of "base physical quantity" in a separate article.

Sometimes these symbols are called logical operators, and sometimes - radicals to emphasize that these symbols do not designate physical quantities. These are the same operators as div, rot and ∇ (nabla) to vector analysis, as conditional icons that represent the logical operations in the Boolean algebra (formal logic), as a differential operator s = d/dt, replacing the suspended operation of differentiation, etc.

To quote also the following passage translation of brochures SI8 (2006, 106) on the dimension: "In general, the dimension of any quantity Q is written as a product of dimensions

dim Q = LαMβTγIδΘεNζJη ,

where the exponents α,β,γ,δ,ε,ζ and η are usually small integers that can be positive, negative or zero, they are called exponents of dimensions". The expression as a product of dimensions symbols, some of which are involuted in the degree called also dimensions formula.

It should also pay attention to what font is used for symbol writer. In designating a scalar physical quantities used italic, in designating a vector physical quantities used straight bold, in the designation of operators (for example, dim, ln, sin) and numeric values ​​(for example, indicators of degree) used direct straight non bold font.

3. Аdvantage of application of dimensions of physical quantities

The dimension of volume of any body (gas, liquid or solid, very small or very large) will always be denoted by L3, independently of any numerical factor standing in the equation to calculate the volume. Similarly, the dimension of the speed of the turtles or the spacecraft is denoted LT-1, although the values ​​of these rates are not comparable. Even if these speeds are measured in different units, such as m/h or km/s.

Let any other inhabited planet local scientists will designate length and time by other symbols. Replace these symbols on the L and T is not difficult. But the dimensions of volume and of speed on the Earth and on another planet would be the same, except that a different recording. And as the units of volume and speed on another planet will certainly be different.

Another example. The symbol of the dimension of mass is M. Therefore, the dimension of such quantity as the force will be referred to SI an expression of MLT-2, which corresponds to SI unit kg·m/s2, which is called the abbreviated Newton (N).

However, as pointed out by A.Vlasov and B.Murin (1990), "dimension, being a quality characteristic of a physical quantity is by no means a complete and exhaustive, but only its conditional characteristic". This observation is very important because a lot of physicists in the last half-century trying to ascribe to dimensions some mystical value that specifies the location of the physical quantity in the systems of quantities and systems of units. Such a direction is described in an article on works R.O. di Bartini and subsequent articles on his followers.

4. Dimensional analysis as a method of validation of laws

In physics, there is one extremely useful mathematical procedure called dimensional analysis.

Suppose we are interested in, is correctly written the equation of the second law of Newton F = ma. We compare the dimensions of the left and right side of the equation. In our example, the dimension of force the left side is equal to the MLT-2, and the dimension the right side consists of the dimension of mass M and the dimension of speed LT-2. The exponents of the dimensions of all three symbols (M, L and T) in both sides of the equation are aligned. So, necessary condition the correctness of recording of the second law of Newton confirmed.

Dimensional analysis is widely used in physics to analyze the equations that are not so simple as F = ma, and for which there is doubt on whether they are true. If the exponents of at least one dimension not match, then it would be an absolute guarantee that the equation is not true.

But if the exponents of dimensions are the same, does that guarantee that the equation is correct? Unfortunately, no. The dimensional analysis is to identify the possible infidelity of record of the equation, but to prove loyalty he can not. Equality degrees in dimensions in left and right sides of the equation is a necessary condition fidelity recording of the equation, but, unfortunately, not sufficient condition. Because in the equation may be present such factors as the quantities, the dimensions of which is equal to 1 because the exponents in the formula of the dimension ​​are 0. In the analysis of dimensions these factors do not affect, but they can significantly affect the physical content of the equation.

What is this quantities with dimension equal to 1? First, the numerical coefficients of proportionality. The number in mathematics of dimension does not have. Its dimension in the analysis of relatively equal 1. But the numerical coefficients in physical laws significantly affect the physical content of the laws. Second, in physics a lot of dimensionless quantities, which is also the dimension is 1. They significantly affect the physical content of the equations, and consider them devoted separate article.

One more note: identity of dimensions of two different physical quantities in some systems of units do not always guarantee the adequacy of the physical content of these quantities . The identity of the physical quantities or lack clarified only when comparing the equations that determine these quantities ​​and called defining equations or coupling equations. You can give an example: in the SI system energy and torque have the same dimension, but they are two different physical quantities. Systems of quantities or systems of units in which this phenomenon may be, require correction. In proposed by the author Energodynamic system of quantities and concepts ESQC similar is not present, because changed set of base quantities ​​in comparison with the set of base quantities in the system ISQ.

In principle the dimensional analysis of physical quantities can be replaced by of units analysis of these quantities, the effect is the same. Very many physicists, and engineers especially, are more used to analyze the units than the dimensional analysis. In the SI at the of units analysis even more features than the of dimensional analysis. The fact that SI are present the units of flat and solid angles, where in dimensions formula of physical quantities unfortunately be absent. And for nothing, as will be shown in the article on rotation angle which should be the basic physical quantity.

Advantages of using of dimensions is not limited to of dimensional analysis. Most are the benefits of the use of dimensions is by systematization of physical quantities, of this can be seen by looking at Tables of physical constants to SI and ESQC and a schemas of layout of physical quantities A.Chuev (2007).

5. Dimensional formula for the generalized coordinates of state

Generalized quantity Q from the equation (1) has long been known in physics. Its called "generalized coordinate" introduced in 1788 by French physicist J.Lagrange to refer to the generalized mechanical quantity obtained in each form of mechanical motion specific content. In this sense, this concept is now used in mechanics (S.Targ, 1995).

200 years after this A.Veynik (1968) came to the conclusion that there is independent from each other and are not reducible to each other elementary forms of movement, each of which is uniquely determined by the physical quantity which A.Veynik called charge. But, in fact, this is a generalized coordinate of J.Lagrange applied not only to assess the state of the mechanical system, and to assess the state of any physical system.

The author of this article (1993) concluded that the term "charge" is a generalized concept not for forms of motion but for forms of physical field and returned in work (1998) to the term "generalized coordinate state". They were asked to assign the dimension of generalized coordinate state Q generalized symbol K (dim Q = K). Content of the symbol K is revealed through the base physical quantities in each form of motion. Subsequently, a systematic approach has resulted in the works (1998, 2004) to the conclusion about existence of generalized physical system. Generalized coordinate state exists only in a generalized physical system.

Introduction of the generalized coordinate is methodical technique convenient by systematization of physical quantities. Dimension K does not introduce additional difficulties in the practice using or dimensions or in the process of dimensional analysis because in the formulas dimension the quantities ​​in specific forms of movement this dimension anymore not. For example, in the form of rectilinear motion K = L. In modern physics, there is another generalized coordinate state characterizing physical field. This electric charge, the dimension of which can be denoted by the symbol Q. The system quantities ISQ instead of electric charge electric current is applied with the dimension I.

6. Lack of definition of dimension

The use of the words "as a product" in the definition of dimension can not be considered good. Such words in conjunction with the record as a product lead to the idea that the symbols of dimensions multiplied. Applies even a term such as "metrological multiplication" (I.Yohansson, 2010). However, after that there is a detailed explanation than the metrological multiplication is different from the arithmetic multiplication, but even is harmful use of such terms as "metrological multiplication". In our view, it would be correct to withdraw from the definition of dimension the words "as a product" and replacing them with the words "as a sequential write symbols of dimensions of basic quantities", pointing out that such a sequence is defined by the standard.

There is no from the metrological multiplication or from the metrological division. But to explain the inconsistency of such a record of units it is possible to carry out in high school when we first met with the units.


1. JCGM 200:2012 International vocabulary of metrology – Basic and general concepts and associated terms (VIM). 3rd ed. 2008 version with minor corrections. URL: http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf,
2. The International System of Units (SI), 8-th edn, 2006, Paris: Bureau International des Poids et Mesures) http://www.bipm.org/en/si/si brochure/
3. I. Johansson, 2010, Metrological thinking needs the notions of parametric quantities, units and dimensions. Metrologia, 47, р.р. 219–230
4. Вейник А.И., 1968, Термодинамика. 3-е изд. – Минск, Вышейшая школа, 464 с. (Veinik A.I., 1968, Thermodynamics. 3rd ed. - Minsk, Vysheyshaya school, 464.)
5. Власов А.Д., Мурин Б.П., 1990, Единицы физических величин в науке и технике. – М., Энергоатомиздат, 176 с. (Vlasov A.D., Murin B.P., 1990, Units of physical quantities in science and technology. - M., Energoatomizdat, 176 pp.)
6. Коган И.Ш., 1993, Основы техники. Киров, КГПИ, 231 с. (Kogan J.Sh., 1993, Fundamentals of Engineering. Kirov, KGPI, 231 p.)
7. Коган И.Ш., 1998, О возможном принципе систематизации физических величин. – “Законодательная и прикладная метрология”, 5, с.с. 30-43. (Kogan J.Sh., 1998, Possible principle of systematization of physical quantities. - "Legal and Applied Metrology", 5, 30-43.)
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9. Тарг С.М., 1995, Краткий курс теоретической механики. – М.: Высшая школа, 334 с. (Targ S.M., 1995, Short course of theoretical mechanics. - M.: High School, 334.)
10. Чуев А.С., 2007, Система физических величин. Текстовая часть электронного учебного пособия. (Chuev А.S., 2007, System of physical quantities.) http://www.chuev.narod.ru/

© J. Kogan Date of the first publication 21.02.2009
Date of last updating 22.01.2014

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