Aijrstem15 740

American International Journal of Research in Science, Technology, Engineering & Mathematics

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ISSN (Print): 2328-3491, ISSN (Online): 2328-3580, ISSN (CD-ROM): 2328-3629 AIJRSTEM is a refereed, indexed, peer-reviewed, multidisciplinary and open access journal published by International Association of Scientific Innovation and Research (IASIR), USA (An Association Unifying the Sciences, Engineering, and Applied Research)

Lagrange Formalism for Electromagnetic Field in Terms of Divisors Dr. Bulikunzira Sylvestre University of Rwanda University Avenue, B.P 117, Butare, Rwanda Abstract: In previous works, with the help of Cartan map, WeylËŠs equation for neutrinos has been written in tensor form, in the form of non-linear Maxwell’s like equations, through complex isotropic vector đ??š = đ??¸âƒ— + ⃗ . It has been proved, that the complex vector đ??š = đ??¸âƒ— + đ?‘–đ??ť ⃗ satisfies non-linear condition đ??š 2 = 0, đ?‘–đ??ť ⃗ 2 = 0 and đ??¸âƒ— . đ??ť ⃗ = 0, obtained by separating real and equivalent to two conditions for real quantities đ??¸âƒ— 2 − đ??ť 2 ⃗ have the same imaginary parts in equality đ??š = 0. Further, it has been proved, that the vectors đ??¸âƒ— đ?‘Žđ?‘›đ?‘‘ đ??ť ⃗ , components of electromagnetic field. For example, under Lorentz relativistic properties as vectors đ??¸âƒ— đ?‘Žđ?‘›đ?‘‘ đ??ť transformations, they are transformed as components of a second rank tensor đ??šđ?œ‡đ?œˆ . In the works that followed, WeylËŠs equation has been written in tensor form, through divisors đ??žđ?œ‡ đ?‘Žđ?‘›đ?‘‘ đ?‘šđ?œ‡ .

In this work, by analogy with WeylËŠs equation, we wrote Maxwell’s equations through divisors đ??žđ?œ‡ đ?‘Žđ?‘›đ?‘‘ đ?‘šđ?œ‡ , and we elaborated the Lagrange formalism for electromagnetic field in terms of divisors đ??žđ?œ‡ đ?‘Žđ?‘›đ?‘‘ đ?‘šđ?œ‡ .

Keywords: Lagrange formalism, electromagnetic field, divisors.

I. Introduction The Lagrange formalism is one of the main tools of the study of the dynamic systems of a vast variety of physical systems, including systems with finite and infinite number of degrees of freedom.

In previous works, via Cartan map, WeylËŠs equation for neutrinos has been written in tensor form, in the form of ⃗⃗ . It has been proved, that the non-linear Maxwell’s like equations, through complex isotropic vector ⃗F = ⃗E + iH ⃗ and H ⃗⃗ have the same properties as vectors E ⃗ and H ⃗⃗ , components of electromagnetic tensor FΟν . For vectors E example, under Lorentz relativistic transformations, they are transformed as components of a second rank tensor FΟν . Further, it has been proved, that the solution of WeylËŠs equation in tensor formalism for free particle as well fulfils Maxwell’s equations for vacuum (with zero at the right side). In the development of the above ideas, in the works that followed, WeylËŠs equation has been written in covariant form, through divisors K Îź and mÎź . By analogy with WeylËŠs equation, Maxwell’s equations also have been written through divisors K Îź and mÎź .

In this work, we shall elaborate Lagrange formalism for electromagnetic field in terms of divisors K Îź and mÎź .

II. Research Method In previous works, WeylËŠs equation for neutrinos has been written in tensor form, in the form of non-linear ⃗ =E ⃗ + iH ⃗⃗ . In the development of these ideas, WeylËŠs Maxwell’s like equations, through isotropic vector F equation for neutrino has been written through divisors K Îź and mÎź . In this work, using the same mathematical method of tensor analysis, we shall write Maxwell’s equations through divisors K Îź and mÎź and we shall elaborate the Lagrange formalism for electromagnetic field in terms of divisors K Îź and mÎź .

A. Maxwell’s Equations in Terms of Divisors In previous works, WeylËŠs equation for neutrinos p0 Ξ = (p âƒ—Ďƒ ⃗ )Ξ, (1) has been written in tensor form as follows ⃗ = iD ⃗⃗ Ă— F ⃗ − (D ⃗⃗ Fi )vi , D0 F (2) ⃗ ⃗ ⃗ ⃗ where F = E + iH. Here â„? ∂ D0 = i , 2 ∂t

⃗D ⃗ = −i � ⃗∇, ⃗ = v

2 ⃗ Ă—H ⃗⃗ E ⃗E2

(3)

.

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Bulikunzira Sylvestre, American International Journal of Research in Science, Technology, Engineering & Mathematics, 12(1), SeptemberNovember, 2015, pp. 33-35

For simplicity, in the following we shall use the natural system of units in which c=â„?=1.

Separating real and imaginary parts in equation (2), we obtain a system of non-linear Maxwell’s like equations ⃗⃗ ⃗ + ∂H = vi (∇ ⃗ Hi ) rot E ∂t { .

(4) ⃗ ⃗ − ∂E = −vi (∇ ⃗ Ei ) rot ⃗H ∂t In terms of isotropic vectors, Maxwell’s equations for vacuum take the form ⃗⃗ Ă— ⃗F D ⃗F = iD { 0 .

(5) ⃗⃗ F ⃗ =0 D ⃗ 2 = 0, However, in general the solution of Maxwell’s equations does not satisfy non-linear isotropic condition F whereas the solution of neutrino equation (2) always satisfies this condition.

Let us introduce bivector field đ?”˝ÎźÎ˝ = K Îź Ë„mν , (6) satisfying isotropic condition ĚƒÎźÎ˝ = 0. đ?”˝ÎźÎ˝ đ?”˝ (7) Here Kđ?œ‡ is a real four vector; mÎź is a complex four vector. In formula (7), the sign "Ë„", means antisymmetrization over indices đ?œ‡ and ν. ⃗ and H ⃗⃗ as follows The four vectors K Îź and mÎź are called divisors and are expressed through E ⃗ |, K Îź = (|E

⃗ Ă—H ⃗⃗ E ), ⃗E2

mÎź = (0,

⃗ F ). ⃗| |E

(8)

Equations (4) written through divisors take the form Dν đ?”˝ÎźÎ˝ = K ν âˆ‚Îź mν .

Maxwell’s equations (5) written through divisors take the form Dν �Ον = 0.

(9) (10)

B. Lagrangian of Electromagnetic Field in Terms of Divisors In spinor formalism, WeylËŠs equation for neutrinos (1) can be obtained by variation principle from Lagrange function i L = (ÎžĎƒÎź âˆ‚Îź Ξ∗ − âˆ‚Îź ÎžĎƒÎź Ξ∗ ).

(11) 2 ⃗ =E ⃗ + iH ⃗⃗ , formula (11) takes the form Written through isotropic vectors F ⠄ ⃗⃗∗ 1 2

i ⃗ − iD ⃗⃗ Ă— F ⃗ + vi (D ⃗⃗ Fi )]F ⃗ ∗ − [D0 F ⃗ ∗ + iD ⃗⃗ Ă— F ⃗ ∗ + vi (D ⃗⃗ Fi∗ )]F ⃗ }â „(FF ) .

L = {[D0 F (12) 2 2 From Lagrange function (12), we immediately obtain the Lagrange function for electromagnetic field in the form i

⃗ − iD ⃗⃗ Ă— F ⃗ ]F ⃗ ∗ − [D0 F ⃗ ∗ + iD ⃗⃗ Ă— F ⃗ ∗ ]F ⃗ }â „( L = {[D0 F 2

⠄ ⃗F ⃗∗ 1 2 F

2 ⠄ ⃗F⃗F∗ 1 2

)

.

When calculating the variations, the quantity ( ) is considered as a constant.

2 It is easy to prove that, the Lagrange function (13), written through divisors takes the form L = (Dν đ?”˝ÎźÎ˝ )mÎźâˆ— − (Dν đ?”˝âˆ—Ον )mÎź .

(13)

(14)

C. Fundamental Dynamical Variables Using Noether's theorem, we can derive from lagrangian (14) expressions for fundamental dynamical physical quantities.

Energy is determined by the formula E = âˆŤ T 00 d3 x, (15) where ∂L ∂L T 00 = mν , 0 + ∗ mÎ˝âˆ— , 0 .

(16) ∂mν ,0

∂mν ,0

Replacing expression (14) in formula (16), we find i

∂m

∂m∗

T 00 = (mÎ˝âˆ— ν − mν ν ).

4 ∂t ∂t Considering the solution for free particle in the form of plane waves mν = [0,

⃗ 0 +iH ⃗⃗ 0 ) (E ⃗ e−2ikt+2ikr⃗ ], ⃗| |E

we find ⃗ |.

T 00 = k|E Similarly for momentum, we have P j = âˆŤ T 0j d3 x,

AIJRSTEM 15-740; Š 2015, AIJRSTEM All Rights Reserved

(17) (18) (19) (20)

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Bulikunzira Sylvestre, American International Journal of Research in Science, Technology, Engineering & Mathematics, 12(1), SeptemberNovember, 2015, pp. 33-35

where i T 0j = [mν ⃗∇j m∗ν − mν∗ ⃗∇j mν ].

4 Replacing expression (18) in formula (21), we obtain ⃗ |.

T 0j = k j |E For charge, we find Q = ∫ j0 d3 x, where ∂L ∂L j0 = i (mν∗ ∗ − mν ). ∂mν ,0

∂mν ,0

Using expression (14), we obtain j0 = mν m∗ν + m∗ν mν .

Replacing expression (18) in formula (25), we get ⃗ |. j0 = |E In the same way, the density of the spin pseudo vector is determined by the formula 1 0 sk = εklm slm , 2 where 0 slm = [−

∂L ∂mi ,0

j

mj Ai,lm −

∂L ∂m∗i ,0

j

mj∗ Ai,lm ].

(21) (22) (23) (24) (25) (26) (27) (28)

Here j

Ai,lm = g il δjm − g im δil .

Replacing expressions (14) in formula (28), we obtain s=

⃗E×H ⃗​⃗ . ⃗| |E

(29) (30)

III. Discussion and Conclusion In this work, we investigated Maxwell’s equations. It is known that, Maxwell’s equations can be written in different forms. Here we proved that, they can also be written through divisors. Similarly with neutrinos field, we obtained the Lagrange function for electromagnetic field in terms of divisors. Using Noether's theorem we derived expressions for fundamental dynamical variables (energy, momentum, charge and spin) conserved in time.

References [1].

[2].

[3].

[4].

[5].

S.Bulikunzira, “Lagrange formalism for neutrino field in terms of complex isotropic vectors,” Asian Journal of Fuzzy and Applied Mathematics, vol.3, no4, Aug. 2015, pp.126-130.

S.Bulikunzira, “Tensor formulation of Dirac equation through divisors,” Asian Journal of Fuzzy and Applied Mathematics, vol.2, no6, Dec. 2014, pp.195-197.

S.Bulikunzira, “Tensor formulation of Dirac equation in standard representation,” Asian Journal of Fuzzy and Applied Mathematics, vol.2, no6, Dec. 2014, pp.203-208.

F.Reifler, “Vector wave equation for neutrinos,” Journal of mathematical physics, vol.25, no4, 1984, pp.1088-1092.

P.Sommers, “Space spinors,” Journal of mathematical physics, vol.21, no10, 1980, pp.2567-2571.

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