- System of Physical Quantities
**System of Physical Quantities of Nikolay A. Plotnikov (SPQ)**— the classification of physical quantities or physical operators, that makes it possible to reveal their dependence on thegeometry ofspace-time andfundamental physical constant s in the form ofdifferential equation s. The system was developed byRussia n physicistNikolai Аleksandrovich Plotnikov during1972 –1978 on the basis of generalphysical laws and is their graphic expression.**History of SPQ**In 17th century,

René Descartes defines and creates the method of 3-Dimensional system with solid axes "X", "Y", "Z". On this basis, he developed principles ofanalytic geometry .In the beginning of 20th century,

Albert Einstein proposed to add to the solid axes system a time axis "t".Around

1930 ,Gabriel Kron [8] defines his theory and writes the book "Tensor Nets Analysis". Kron studies nets of electric machinery and tries to use Tensor Analysis and up-to-date topology achievements. Though models of electric machinery are considered in Kron's works, Kron mentions the possibility of application of this kind of actuarial mathematics for computation of other kinds of physical systems.In the middle of 20th century, scientists searched for systematization of decrees of nature in space and time solid axes system. Results were the placement of Physical Quantities of mechanics and gravitation into

SGS system without taking into account mathematical analysis of field and possibility of application of one mathematical model for some physical processes of different kinds.Such well-known scientists and engineers were O de Bartini and

Kusnezov [6,7] . These studies led them to the kinematic system of Physical Quantities proposed by O .de Bartini. This kinematic system of Physical Quantities uses as base dimensional units only two: length [L] and time [T] . All other physical units including mass are considered derivative from these two base units. They are expressed in derived units [L] and [T] .O de Bartini system describes physical system models. From my point of view Kron's works have proposals of mathematical description of physical systems.

In

1978 Plotnikov Nikolay [5] published System of Physical Quantities (SPQ) invented by him. It is based onSI units system. The SPQ uses space and time solid axes system and additional axis of Fundamental Physical Constants /FPC/In

1981 , in his article,Deschamps [4] published two graphs (DAG) for electromagnetic differential forms and description of differential-form quantities. The both Deschamps' graphs for electromagnetic differential forms are included in the SPQ of Plotnikov.Stokes' theorem andGauss' theorem and also differential forms of different physical dimentions are described in Plotnikov's publication.In

2004 ,Ismo V. Lindell [1] published the book with detailed description of differential-form quantities and its application to EM theory. This monograph is an excellent and deep introduction to modern language of EM theory. Ismo V. Lindell's book includes last results of studies of different media (for example: bi-anisotropic). Ismo V. Lindell developed mathematical formalism of physical processes of EM field.Lately, due to development of computing systems, it is getting more actual to use differential-form theory for description and computing methods [2,3] . There are a lot scientific publication on this theme. This direction of mathematical physics and computer modeling is developing very fast.

Already in

1978 Russia n scientistPlotnikov Nikolay Alexandrovich published the results of his own long-term researches in this field. Unfortunately Plotnikov N.A. passed away two years ago. But his achievements are saved due to his works. So far his heritage is not known in our country and abroad. From my point of view his works for modern physics is very actual because modern researchers are only discovering these directions of physics, which already are described and studied by Plotnikov N.A.SPQ of Plotnikov is not a system of physical units (like

SI ,SGS and etc.) But SPQ is based onSI . System of Physical Quantities provides structural scheme of interrelations between different physical quantities in terms of mathematical expressions given bydifferential forms and etc.**Theory of SPQ****Deschamps graph**In

1981 in his articleDeschamps [4] published two graphs (DAG) forelectromagnetic differential form s.:$egin\{matrix\}0-forms:phi\backslash \{Bigdownarrow\}^\{-d\}\backslash 1-forms:Alongrightarrow^\{-dt\}EH\backslash \{Bigdownarrow\}^\{d\}\{Bigdownarrow\}^\{-d\}\{Bigdownarrow\}^\{d\}\backslash 2-forms:Blongrightarrow^\{-dt\}0Dlongrightarrow^\{-dt\}J\backslash \{Bigdownarrow\}^\{d\}\{Bigdownarrow\}^\{d\}\{Bigdownarrow\}^\{d\}\backslash 3-forms:0\; holongrightarrow^\{-dt\}0\backslash end\{matrix\}$Maxwell -Faraday and Maxwell-Ampère equations (But the most common modern notation for these equations was developed byOliver Heaviside ) as aDeschamps graph for electromagnetic differential forms quantities in three dimensions.**Base of SPQ**The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε

_{0}and μ_{0}.Magnetic field and magnetic flux density equation: :$mathbf\{H\}\; =\; 1\; /\; mu\_0\; cdot\; mathbf\{B\}$Electric field andelectric displacement field equation::$mathbf\{E\}\; cdot\; varepsilon\_0\; =\; mathbf\{D\}$Velocity of light c::$c\; =\; frac\{1\}\{sqrt\{mu\_0\; varepsilon\_0$or:$\{1\; over\; mu\_0\; \}\; =\; varepsilon\_0\; c^2$Vacuum impedance is:$\{Z\_0\; =\; mu\_0\; ,\; c^1\}$or:$\{1\; over\; R\_0\; \}\; =\; varepsilon\_0\; c^1$Replace ε_{0}and μ_{0}constants together with the signs equally by the appropriate direction arrows and obtain system relations between theelectromagnetic field one and two-forms:Each arrow unambiguously corresponds to the physical constant, which to find on one horizontal line with the arrow. Arrows are directed to the side of sign equally for the appropriate expressions. Horizontal arrows are equivalent to (dual)Hodge star operator * for the electrical ormagnetic field . The mappings defined by various mediumdyadic s. Expression with impedance of free space relates to the vertical arrows.**Additional vertical (columns) and horizontal (line) graphs**The line number 1 is the list of physical quantities names of vertical graphs (columns).

The line number 2 is the field characteristics line and

differential form numbers (denominator).The line number 3 is the list of the physical quantities results of the multiplication of symmetrical physical quantities relative to the vertical line.

The vertical graphs A (column A) is the fundamental fhysical constants graphs.

The vertical graphs B (column B) is the

matter forms graphs (Elasticity (physics) ,Magnetostatics ,Electrostatics etc.).**Poincaré's lemma and de Rham's theorems**Following physical formulas,

Poincaré's lemma andde Rham theorem , we unroll the System of Physical Quantities of Nikolay A. Plotnikov.Magnetic

energy density ormagnetic pressure ::$old\{P\}\; =\; old\{H\}\; cdot\; old\{B\}\; =\; |old\{B\}|^2\; \{1\; over\; mu\_0\; \}\; =\; |old\{H\}|^2\; mu\_0$Electricenergy density orelectric pressure ::$old\{P\}\; =\; old\{E\}\; cdot\; old\{D\}\; =\; |old\{D\}|^2\; \{1\; over\; varepsilon\_0\; \}\; =\; |old\{E\}|^2\; varepsilon\_0$Let us multiply and let us divide the right sides of the expressions on

space physical quantity $old\{l\}$ ::$old\{P\}\; =\; old\{l\}\; ,\; old\{H\}\; cdot\; \{1\; over\; old\{l\; ,\; old\{B\}$:$old\{P\}\; =\; old\{l\}\; ,\; old\{E\}\; cdot\; \{1\; over\; old\{l\; ,\; old\{D\}$The

electric potential orscalar potential is::$-\; old\{d\}\; phi\; =\; old\{E\}$or the same::$-\; vec\{\; abla\}\; phi\; =\; vec\{old\{E$The

electric charge density is:$ho\_e\; =\; old\{d\}\; wedge\; old\{D\}$or the same::$ho\_e\; =\; vec\{\; abla\}\; cdot\; vec\{old\{D$The

magnetic tension or the magneticpotential difference ::$old\{V\}\; =\; old\{l\}\; ,\; old\{H\}$The

magnetic source equal to zero (Magnetic monopole ) ::$operatorname\{div\}\; old\{B\}\; =\; 0$Place to the SPQ the analytical expressions in the graphic form. Operation with the space physical quantity $old\{l\}$ (

wedge product ) replace by the graphic, short, horizontal pointer. The direction of pointer determine to the side of equal sign. This operation type put to the top of the SPQ (line "4").**List of the physical processes****Physical processes of the magnetism**: $vec\{\; abla\}\; imes\; vec\{H\}\; =\; vec\{j\}$

: $vec\{\; abla\}\; imes\; vec\{A\}\; =\; vec\{B\}$

: $int\_S\; vec\{B\}\; ,\; \{\; m\; d\}vec\{S\}\; =\; Phi\_B$

: $vec\{P\_m\}\; =\; i\; ,vec\{S\}\; ,\; vec\{n\}$

**Physical processes of the electrostatics**: $N\; =\; oint\_S\; vec\{E\}\; ,\; \{\; m\; d\}vec\{S\}$

: $Q\; =\; int\_V\; ho\; ,\; \{\; m\; d\}vec\{V\}$

: $vec\{\; au\_q\}\; =\; \{Q\; over\; l\}\; ,\; vec\{n\}$

: $-\; ho\; /\; varepsilon\_0\; =\; vec\{\; abla^2\}\; ,\; phi$

**Analytical expressions for the relation in the time of the physical processes****Use of the Differential Forms in Electromagnetics****Maxwell's equations in terms of differential forms in a 3-dimensional space****Maxwell's equations in terms of differential forms in a 4-dimensional space-time**space-time In additional horizontal line 2 the numbers of

differential form s for the three-dimensional space are shown.:$au\; =\; c\; ,\; t$ :$old\; d\; au\; partial\_t\; =\; \{1\; over\; c\}\; ,\; partial\_t\; ,$:$c\; =\; frac\{1\}\{sqrt\{mu\_0\; varepsilon\_0$ :$Tau\; =\; old\; t\; +\; old\; l\; wedge\; old\; d\; au$

:$vec\{vec\{I\_E^T\; =\; sum\_\{i=1\}^3\; varepsilon\_i\; old\; e\_i$:$Alpha\; =\; omega\; +\; v\; wedge\; old\; d\; au\; =\; vec\{vec\{I\_E^\{(2)T\}\; +\; vec\{vec$I_E^T}_wedge^wedge , old d au , old e_ au = {1 over 2} sum_{i=1}^3 sum_{j=1}^3 (varepsilon_i wedge varepsilon_j )(old e_i wedge old e_j) + sum_{i=1}^3 ( varepsilon_i wedge old d au ),( old e_i wedge old e_ au )

:$old\; d\; =\; old\; d\_E\; +\; old\; d\; au\; ,\; partial\_\; au$:$old\; d\_E\; =\; sum\_\{i=1\}^3\; old\; d\; x\_i\; ,\; partial\_\{x\_i\}\; =\; old\; d\; x\; ,partial\_x\; +\; ,old\; d\; y\; ,partial\_y\; +\; old\; d\; z\; ,partial\_z$.

:$Phi\; =\; B\; +\; E\; wedge\; old\; d\; au$

:$old\; d\; wedge\; Phi\; =\; 0$

:$gamma\; =\; ho\; -\; J\; wedge\; old\; d\; au$

:$Psi\; =\; D\; +\; H\; wedge\; old\; d\; au$

:$old\; d\; wedge\; Psi\; =\; gamma\_e$

:$Phi\; =\; old\; d\; wedge\; alpha\; =\; old\; d\; wedge\; (\; alpha\; +\; old\; d\; psi\; )$

:$alpha\; =\; A\; -\; phi\; old\; d\; au$

:$alpha\; o^d\; Phi\; o^d\; 0$:$Psi\; o^d\; gamma\; o^d\; 0$

:$Psi\; =\; vec\{vec\{M\; ,\; |\; ,\; Phi$

**References*** [1. Lindell I. V., Differential Forms in Electromagnetics. IEEE Press, Wiley Interscience, 2004]

* [2. http://www.llnl.gov/CASC/emsolve/pdf/cmes-paper.pdf P. Castillo, J. Koning, R. Rieben and D. White, A Discrete differential forms framework for computational electromagnetism, Computer Modeling in Engineering and Sciences, 2004, Vol 5, No 4, pp. 331-346.]

* [3. http://www.llnl.gov/CASC/emsolve/pdf/241640.pdf Castillo P., Koning J., Rieben R., Stowell M., White D., Discrete Differential Forms: A Novel Methodology for Robust Computational Electromagnetics. California, Lawrence Livermore National Laboratory Technical Information Department's Digital Library, January 17,2003]

* [4. Deschamps G., Electromagnetics and differential forms. IEEE Proceedings, Vol. 69, No. 6, pp. 676-687. 1981.]

* [5. http://plotnikovna.narod.ru/ Плотников Н.А. Система физических величин. ВОИР и Вологодский Областной Совет ВОИР. Вологда. 1978., (ББК 22.3 с, УДК 53.081)]

* [6. Кузнецов П. Г. Искусственный интеллект и разум человеческой популяции. - В кн.: Александров Е. А. Основы теории эвристических решений. - М., 1975.]

* [7. Бартини Р. О., Кузнецов П. Г. Множественность геометрий и множественность физик. - В сб.: Моделирование динамических систем". Брянск, 1974, с. 18-29.]

* [8. Kron G. Tensor Analysis of Networks. N. Y., 1939.]

* [9. http://www.chuev.narod.ru/ Anatoly Chuev-персональный сайт]

* [10. Чуев А. С. Физическая картина мира в размерности "длина-время" М., СИНТЕГ, 1999 г. с. 96]

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