Available online Journal of Scientific and Engineering Research, 2016, 3(6): Research Article

Transcription

1 Aailable online 06, 3(6):70-74 Research Article ISSN: CODEN(USA): JSERBR Second Order Lagrangian Hassaballa M Abdalgadir *, Mubarak Dirar, Zoalnoon A Abeid Allah, Lutfi M Abdalgadir Sudan Uniersity of science & technology, Khartoum, Sudan Al Zaeem Al Azhary Uniersity, Bahry, Khartoum, Sudan Abstract Recently a Lagrangian dependent on the second order deriaties was shown to be successful in describing and soling long standing graitational problems. This gies motiation to construct second order electromagratic Lagrangian, which may hopefully can open a new horizon to sole disasters electromagnetic problems like self charge energy problem. In this work second order electromagnetic Lagrangian was deried with acuum current density terms. This Lagrangian is used to derie a new Maxwell's equations, which predicts that both electric charges and acuum can generate photons. The new Hamiltonian consists of an additional terms representing source emitting or obsering photons beside acuum energy terms. The equation of motion reduces to the ordinary equation of motion and to ordinary Hamiltonian with additional source term in the absence of acuum energy term. Keywords Second order Lagrangian, electromagnetic field, Vacuum current density, emitting and absorbing term. Introduction Electromagnetic waes (E.M.W) plays an important role in our day life now. This is since it is widely used in telecommunications and multimedia. They are described by Maxwell's equations. Maxwell's equations (M. E) unify electric and magnetic phenomena [-]. The M. E are deried from the laws of electricity and magnetism [3]. But later on, Lagrange formalism is utilized to derie them. Despite the remarkable success of this formalism, it suffers from some drawbacks. For instance, the Lagrange formalism, fails in describing some graitational phenomena, namely that concerns the graitational energy and the early unierse [4]. Maxwell's equations also does not able to find finite charge self energy. Recently, attempts were made to use a generalized principle of least action to cure these defects. In this generalized ersion the Lagrangian consists of an additional terms that depends on the second deriaties of the generalized coordinates [4-6]. Motiated by the success of the generalized Lagrangian in most soling graitational problems, the generalized Lagrangian is utilized. In this work one deries (M. E) [7-0]. This is done in section (3), sections (4) and (5) are deoted for discussion and conclusion.. Ordinary Lagrangian of Electromagnetic Field The electromagnetic field Lagrangian is gien by A L = 8π c T + φ 8π A = 8π = oa i i A o 8π ia o i A o 70

2 Abdalgadir HM et al, 06, 3(6):70-74 Px = Px = x q = x A 4πc P = A 4πc + φ = 4πc 0A i i A o Where A i 's stands for magnetic potential, while φ represents the electric potential. The corresponding Hamiltonian is giens by H = P A t L = A 4π + φ A L = A 4π + φ A A 8π + φ A A 8π 8π A = A 8π + φ. A A 8π 8π A = A 8π + φ. A + φ A 4π 8π A = π C P CP. φ + 8π A The equation of motion and the Hamiltonians are deried from Lagrangian dependent on the field ariables and their first deriaties. 3. New second order Lagrangian: The equation of motion for second order Lagrangian takes the from A μ u μ A u The corresponding Hamiltonian is also gies by c A x t + Q x (.) + ςμ ςμ A = 0 (3.) T 0 0 = H = o A o A i ς + i 0ς A i oς A oς A i L (3.) i Where i is a dummy indices, beside ς. The appropriate second order lagrangian that can gie Maxwell's equations and the correct Hamiltonian gien by: L = c η ρς ρ A ς ρ A ς c A λ J ψ λ + J λ J ψ J A J ψ λ = qψ γ λ Ψ = charge current density J A = c 3 λρ A ρ ρρ A λ = surce generating or absorbing field (3.3) J μ u acuum current density thus the system of matter hae charge current density corresponding to rest mass energy which at the same time acts as a source emitting or absorbing field mediators. to find the equation of motion of electromagnetic field (E. M. F) one differentiate w. r. t A to get: = c A J λ λ ψ + J (3.5) Also = c μ A η μ μ A μ A A μ + η μ A μ μ A = c η μ μ A A μ + η μ A μ μ A = 4c η μ μ A A μ (3.6) Equations (5) and (6) can help in finding the equation of motion. To find the Hamiltonian, one needs to differentiate L w. r. t. to time deriatie of magnetic potential A i to get. o A i o A i = 7

3 Abdalgadir HM et al, 06, 3(6):70-74 By choosing: = o A i c η oi o A i i A o + c η io i A o o A i o A i c η oi o A i i A o + i A o o A i ( ) o A i = c o A i i A o o A i (3.7) o = x 0 x 0 = ict A 0 = iφ x = x, x = y x 3 = z A = A x A = Ay A 3 = Az (3.8) i =,,3 One gets. o A o A i = c j A x φ j i x j A x + = c A x + φ A x x = c A + φ A The terms differentiated with respect to the second order deriaties are gien according to equation (3.3) and (3.4) to be. = c λgg μ A μ A λg A g λg A λ Let : μ = λ = o, ς = = g = i, μ = ς = g = 0 = λ = i = c μςς μς A μς A ςς A μ = c μςς μς A μ A μ μς A = c μςς c ςς = c 3 c 3 = 0 (3.8) Thus according to equations (3-), (3-4), (3-5) and (3-6) and (3-8) the equation of motion is gien by μ 4c η μ μ A A μ c J ψ + J = 0 By setting (3.9) One gets the ordinary equation of motion c = 8π c = c μ A μ μμ A = 4π J c ψ + J (3.0) The Hamiltonian can be found by a direct substitution of equations (3.8), (3.7) and (3.8) in equation (3.) to get A H = c + φ A c η oi o A i i A o c η ij i A j i A j +J ψ + J A + c A J ψ + J According to equations (3.8) the Hamiltonian is gien by H = c A + φ A = c i A x φ i i A x φ i x x c A + J ψ + J A + c A J ψ + J c A + φ A 7

4 Abdalgadir HM et al, 06, 3(6): c c = c A + φ = c = 8π A + φ c t A + J ψ + J A + c A J ψ + J (3.) A + φ A + φ φ + A +J ψ + J A + c A J ψ + J A + φ c A t + φ + A +J ψ + J A + c A J ψ + J A + φ A A 8π 6π c p 8πcp. φ + A + J ψ + J A + c A J ψ + J H = πc p cp. φ + A + J 8π ψ + J A + c A J ψ + J (3.3) There c = 8π This Hamiltonian is the usual electromagnetic field Hamiltonian. Discussion This second order Lagrangian was shown to be successful in describing the graitational phenomena by the so called generalized general relatiity. This motiates us to try to construct new second order Lagrangian for the electromagnetic field, as shown by equation (3.3). In this Lagrangian matter energy manifests is self through two terms. The first term J ψ represents energy stored in a charge which equals charge energy per parties γ μ multiplied by the number of particles ψ. The second terms represents the contribution of charged system in generating or obsering photons. This term, J A, which represents the a field source manifests its role in absorbing or generating field through the equation of motion (3.0). Which shows dependence of J A on second order deriatie of A μ. It is ery interesting to note that this terms gies no contribution to the equation of motion according to equations (3.8) and (3.). At the same time the Hamiltonian of equation (3.3) shows the appearance of the terms J A which describes absorption or emission of photons energy by charged systems, beside the terms J ψ which represent the energy stored in electric charges. It is also important to note that equation (3.0) shows that both electric charges and acuum energy can generate electromagnetic field. The generation of e. m. field was proed by Cassimar effect. The Hamiltonian in equations (3.3) too also hae a term recognizing the acuum energy. This is also experimentally erified by Cassimar effect. Conclusion The Lagrangian depending on second order field deriaties shows that acuum energy can generate electromagnetic field as well as electric charges. It shows also that the Hamiltonian consists of terms recognizing charge energy, source energy, and acuum energy. References []. Carlson, John B. (975) "Lodestone Compass: Chinese or Olmec Primacy: Multidisciplinary analysis of an Olmec hematite artifact from San Lorenzo, Veracruz, Mexico", Science, 89 (405 : 5 September), p []. L.H. Greenberg (978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN. [3]. J.B. Marion, W.F. Hornyak (984). Principles of Physics. Holt-Saunders International Saunders College. 73

5 Abdalgadir HM et al, 06, 3(6):70-74 [4]. M. Dirar et al, Int. J of Astronomy end Astrophysics (IJAA), 3, 3-36 (03). [5]. M. Dirar, A. El tahir, M. Shaddad, Mod. Phy. Let. A, V. 3, N. 37 (998). [6]. K. G Elgaylani et al, Int. J. of phy. Sciences, Vol. (), 05-00, Feb. 04 (04). [7]. M. Dirar et al, J of Sc and Tech, V. (6) (000). [8]. H. M. Kamal et al, Sudan. J. Basic science (M), N. 4, (005). [9]. H. M. Kamal et al, Sudan. J. Basic sciences, N, 0, 89-00, (005). [0]. H. M. Kamal et al, Sudan. J. Basic sciences, N, 7, 5-38, (03). []. Z.A.Abeid Allah, Elixir Nuclear & Radiation Phys. 9 (06)