System of Physical Quantities

From Wikipedia, the free encyclopedia

This article is actively undergoing translation from another Wikipedia for a while.

As a courtesy, please do not edit this article while this message is displayed. The person who added this notice will be listed in its edit history should you wish to contact him or her.

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 the geometry of space-time and fundamental physical constants in the form of differential equations. System is developed by Russian physicist Nikolai Аleksandrovich Plotnikov in 1972 - 1978 on the basis of general physical laws and is their graphic expression.

1 History of SPQ
2 Theory of SPQ
3 Use of the Differential Forms in Electromagnetics
- 3.1 Maxwell's equations in terms of differential forms in a 3-dimensional space
- 3.2 Maxwell's equations in terms of differential forms in a 4-dimensional space-time
4 References

[edit] 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 of analytic 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 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 on SI 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 and Gauss' theorem and also differential forms of different physical dementions 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 medias (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.

I should point out that already in 1978 Russian scientist Plotnikov 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 system of physical units (like SI, SGS and etc.) But SPQ is based on SI. System of Physical Quantities provides structural scheme of interrelations between different physical quantities in terms of mathematical expressions given by differential forms and etc.

[edit] Theory of SPQ

[edit] Deschamps graph

In 1981 in his article Deschamps [4] published two graphs (DAG) for electromagnetic differential forms.

$\begin{matrix} 0-forms:&&&\phi&&&&\\ &&&{\Big\downarrow}^{-d}&&&&\\ 1-forms:&A&\longrightarrow^{-dt}&E&&&H&\\ &{\Big\downarrow}^{d}&&{\Big\downarrow}^{-d}&&&{\Big\downarrow}^{d}&\\ 2-forms:&B&\longrightarrow^{-dt}&0&D&\longrightarrow^{-dt}&J\\ &{\Big\downarrow}^{d}&&&{\Big\downarrow}^{d}&&{\Big\downarrow}^{d}&\\ 3-forms:&0&&&\rho&\longrightarrow^{-dt}&0\\ \end{matrix}$

Maxwell-Faraday and Maxwell-Ampère equations (But the most common modern notation for these equations was developed by Oliver Heaviside) as a Deschamps graph for electromagnetic differential forms quantities in three dimensions.

[edit] Base of SPQ

The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε₀ and μ₀.

Magnetic field and magnetic flux density equation:

$\mathbf{H} = 1 / \mu_0 \cdot \mathbf{B}$

Electric field and electric displacement field equation:

$\mathbf{E} \cdot \varepsilon_0 = \mathbf{D}$

Velocity of light c:

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

${1 \over \mu_0 } = \varepsilon_0 c^2$

Vacuum impedance is

${Z_0 = \mu_0 \, c^1}$

${1 \over \R_0 } = \varepsilon_0 c^1$

Replace ε₀ and μ₀ constants together with the signs equally by the appropriate direction arrows and obtain system relations between the electromagnetic field one and two-forms:

$\begin{matrix} {{1 \over \mu_0 } = \varepsilon_0 c^2}&\Big\vert&\vec{H}&\larr&\vec{B}&\Big\vert\\ {{1 \over \R_0 } = \varepsilon_0 c^1}&\uparrow&---&\Big\vert&---&\downarrow\\ {\mathbf{\varepsilon_a \over \varepsilon } = \varepsilon_0 c^0 }&\Big\vert&\vec{E}&\rarr&\vec{D}&\Big\vert\\ \end{matrix}$

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 or magnetic field. The mappings defined by various medium dyadics. Expression with impedance of free space relates to the vertical arrows.

[edit] 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.).

$\begin{matrix} A&\Big\vert&''4''&\Big\vert\bold{d}\wedge\rarr\Big\vert&&&&\Big\vert\bold{d}\wedge\rarr\Big\vert&&\Big\vert ~Matter~forms~ \\ ---&\Big\vert&---&---&---&---&---&---&---&\Big\vert ------- \\ {{1 \over \mu_0 } = \varepsilon_0 c^2}&\Big\vert&V&\Big\vert&\vec{H}&\larr&\vec{B}&\Big\vert & \, & \Big\vert Magnetostatics\\ {{1 \over \R_0 } = \varepsilon_0 c^1}&\Big\vert&---&\uparrow&---&\Big\vert&---&\downarrow&---&\Big\vert ------- \\ {\mathbf{\varepsilon_a \over \varepsilon } = \varepsilon_0 c^0 }&\Big\vert&\phi&\Big\vert&\vec{E}&\rarr&\vec{D}&\Big\vert&\bold{\rho_e}&\Big\vert ~~ Electrostatics \\ ---&\Big\vert&---&---&---&---&---&---&---&\Big\vert ------- \\ 1&\Big\vert&Scalar&\Big\vert&Tenseness&\Big\vert&Induction&\Big\vert&Qantum&\Big\vert ~~~~~~~~~~~~~~~~~~~~~ \\ &\Big\vert&potential&\Big\vert&~~~&\Big\vert&~~~&\Big\vert&density&\Big\vert ~~~~~~~~~~~~~~~~~~~~~ \\ ---&\Big\vert&---&---&---&---&---&---&---&\Big\vert ~~~~~~~~~B~~~~~~~~~~ \\ 2&\Big\vert&{1 \over 0}&\Big\vert&{2 \over 1}&\Big\vert&{1 \over 2}&\Big\vert&{2 \over 3}&\Big\vert ~~~~~~~~~~~~~~~~~~~~~ \\ ---&\Big\vert&---&---&---&---&---&---&---&\Big\vert ~~~~~~~~~~~~~~~~~~~~~ \\ 3&\Big\vert&~~~&~~~&Pressure&P&~~~&~~~&~~~&\Big\vert ~~~~~~~~~~~~~~~~~~~~~ \\ \end{matrix}$

[edit] Poincaré's lemma and de Rham's theorems

Following physical formulas, Poincaré's lemma and de Rham theorem, we unroll the System of Physical Quantities of Nikolay A. Plotnikov.

Magnetic energy density or magnetic pressure:

$\bold{P} = \bold{H} \cdot \bold{B} = |\bold{B}|^2 {1 \over \mu_0 } = |\bold{H}|^2 \mu_0$

Electric energy density or electric pressure:

$\bold{P} = \bold{E} \cdot \bold{D} = |\bold{D}|^2 {1 \over \varepsilon_0 } = |\bold{E}|^2 \varepsilon_0$

Let us multiply and let us divide the right sides of the expressions on space physical quantity $\bold{l}$ :

$\bold{P} = \bold{l} \, \bold{H} \cdot {1 \over \bold{l}} \, \bold{B}$

$\bold{P} = \bold{l} \, \bold{E} \cdot {1 \over \bold{l}} \, \bold{D}$

The electric potential or scalar potential is:

$- \bold{d} \phi = \bold{E}$

or the same:

$- \vec{\nabla} \phi = \vec{\bold{E}}$

The electric charge density is

$\rho_e = \bold{d} \wedge \bold{D}$

or the same:

$\rho_e = \vec{\nabla} \cdot \vec{\bold{D}}$

The magnetic tension or the magnetic potential difference:

$\bold{V} = \bold{l} \, \bold{H}$

The magnetic source equal to zero (Magnetic monopole) :

$\operatorname{div} \bold{B} = 0$

Place to the SPQ the analytical expressions in the graphic form. Operation with the space physical quantity $\bold{l}$ (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").

[edit] List of the physical processes

[edit] Physical processes of the magnetism

$\vec{\nabla} \times \vec{H} = \vec{j}$

$\vec{\nabla} \times \vec{A} = \vec{B}$

$\int_S \vec{B} \, {\rm d}\vec{S} = \Phi_B$

$\vec{P_m} = i \,\vec{S} \, \vec{n}$

[edit] Physical processes of the electrostatics

$N = \oint_S \vec{E} \, {\rm d}\vec{S}$

$Q = \int_V \rho \, {\rm d}\vec{V}$

$\vec{\tau_q} = {Q \over l} \, \vec{n}$

$- \rho / \varepsilon_0 = \vec{\nabla^2} \, \phi$

[edit] Analytical expressions for the relation in the time of the physical processes

[edit] Use of the Differential Forms in Electromagnetics

[edit] Maxwell's equations in terms of differential forms in a 3-dimensional space

[edit] 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 forms for the three-dimensional space are shown.

$\tau = c \, t$

$\bold d \tau \partial_t = {1 \over c} \, \partial_t \,$

$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$

$\Tau = \bold t + \bold l \wedge \bold d \tau$

$\vec{\vec{I_E^T}} = \sum_{i=1}^3 \varepsilon_i \bold e_i$

$\Alpha = \omega + v \wedge \bold d \tau = \vec{\vec{I_E^{(2)T}}} + \vec{\vec{{I_E^T}_\wedge^\wedge}} \, \bold d \tau \, \bold e_\tau = {1 \over 2} \sum_{i=1}^3 \sum_{j=1}^3 (\varepsilon_i \wedge \varepsilon_j )(\bold e_i \wedge \bold e_j) + \sum_{i=1}^3 ( \varepsilon_i \wedge \bold d \tau )\,( \bold e_i \wedge \bold e_\tau )$

$\bold d = \bold d_E + \bold d \tau \, \partial_\tau$

$\bold d_E = \sum_{i=1}^3 \bold d x_i \, \partial_{x_i} = \bold d x \,\partial_x + \,\bold d y \,\partial_y + \bold d z \,\partial_z$ .

$\Phi = B + E \wedge \bold d \tau$

$\bold d \wedge \Phi = 0$

$\gamma = \rho - J \wedge \bold d \tau$

$\Psi = D + H \wedge \bold d \tau$

$\bold d \wedge \Psi = \gamma_e$

$\Phi = \bold d \wedge \alpha = \bold d \wedge ( \alpha + \bold d \psi )$

$\alpha = A - \phi \bold d \tau$

$\alpha \to^d \Phi \to^d 0$

$\Psi \to^d \gamma \to^d 0$

$\Psi = \vec{\vec{M}} \, | \, \Phi$

[edit] 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]

Retrieved from "http://en.wikipedia.org../../../s/y/s/System_of_Physical_Quantities_fad8.html"

Categories: Articles in translation | Physics