Research Article
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Year 2022, , 38 - 43, 30.10.2022
https://doi.org/10.47087/mjm.1160600

Abstract

References

  • O.D. Kichmarenko, L.I. Plotnikova, N.V. Skripnik, Variational Calculus (in Russian), Odessa, Astroprint (2009).
  • A.V. Panteleev, Calculus of Variations in examples and problems (in Russian), Ed. MAI (2000).
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  • J. Lopez-Bonilla, R. Lopez-Vazquez, S. Vidal-Beltran, Linearization of (second order operator)1=2, World Scientic News, 108 (2018) 224-228.
  • B. Mahon, The forgotten genius of Oliver Heaviside, Prometheus Books (2017).
  • P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A117 (1928) 610-624.
  • H. S. Kragh, The genesis of Dirac's relativistic theory of electrons, Archive for History of Exact Sciences 24(1) (1981) 31-67.
  • P. Lam-Estrada, J. Lopez-Bonilla, R. Lopez-Vazquez, On the Noether's Theorem, Prespace-time Journal, 6(4) (2015) 322-325 .

DIRAC'S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT

Year 2022, , 38 - 43, 30.10.2022
https://doi.org/10.47087/mjm.1160600

Abstract

In this paper, we adopt a matrix treatment to solve the variational problem that consists of determining the physical path traveled by light between two points in a medium whose refractive index depends on a spatial coordinate. The considered treatment begins with the trivial repetition of the expression of the value of the considered functional, repetition expressed in the form of a matrix. Next, we adopt the trick (of Dirac) originally used as part of the construction of the dynamic equation of relativistic quantum mechanics, which allows us to rewrite the (now) matrix integrand in the expression of the value of the functional in terms of the sum of two (non-diagonal) matrices brought externally to the problem, which are determined based on some requirements. As a result of this development, we obtain two equivalent versions of Snell's law.

References

  • O.D. Kichmarenko, L.I. Plotnikova, N.V. Skripnik, Variational Calculus (in Russian), Odessa, Astroprint (2009).
  • A.V. Panteleev, Calculus of Variations in examples and problems (in Russian), Ed. MAI (2000).
  • M.L. Krasnov, G.I. Makarenko, A.I. Kiseliov, Calculo Variacional (in Spanish), Ed. Mir-Moscu (1976).
  • J. Chaves, Introduction to Nonimaging Optics, second edition, Taylor & Francis Group (2016).
  • J. Garcia, Como llego Dirac a su ecuacion available on the internet, http://aula141.cat/wp-content/uploads/2015/10/Presentaci%C3%B3n_Dirac.pdf
  • J. Lopez-Bonilla, R. Lopez-Vazquez, S. Vidal-Beltran, Linearization of (second order operator)1=2, World Scientic News, 108 (2018) 224-228.
  • B. Mahon, The forgotten genius of Oliver Heaviside, Prometheus Books (2017).
  • P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A117 (1928) 610-624.
  • H. S. Kragh, The genesis of Dirac's relativistic theory of electrons, Archive for History of Exact Sciences 24(1) (1981) 31-67.
  • P. Lam-Estrada, J. Lopez-Bonilla, R. Lopez-Vazquez, On the Noether's Theorem, Prespace-time Journal, 6(4) (2015) 322-325 .
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Juan Bulnes 0000-0003-4367-0453

José Luıs López-bonılla This is me 0000-0003-3147-7162

Publication Date October 30, 2022
Acceptance Date October 19, 2022
Published in Issue Year 2022

Cite

APA Bulnes, J., & López-bonılla, J. L. (2022). DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT. Maltepe Journal of Mathematics, 4(2), 38-43. https://doi.org/10.47087/mjm.1160600
AMA Bulnes J, López-bonılla JL. DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT. Maltepe Journal of Mathematics. October 2022;4(2):38-43. doi:10.47087/mjm.1160600
Chicago Bulnes, Juan, and José Luıs López-bonılla. “DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT”. Maltepe Journal of Mathematics 4, no. 2 (October 2022): 38-43. https://doi.org/10.47087/mjm.1160600.
EndNote Bulnes J, López-bonılla JL (October 1, 2022) DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT. Maltepe Journal of Mathematics 4 2 38–43.
IEEE J. Bulnes and J. L. López-bonılla, “DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT”, Maltepe Journal of Mathematics, vol. 4, no. 2, pp. 38–43, 2022, doi: 10.47087/mjm.1160600.
ISNAD Bulnes, Juan - López-bonılla, José Luıs. “DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT”. Maltepe Journal of Mathematics 4/2 (October 2022), 38-43. https://doi.org/10.47087/mjm.1160600.
JAMA Bulnes J, López-bonılla JL. DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT. Maltepe Journal of Mathematics. 2022;4:38–43.
MLA Bulnes, Juan and José Luıs López-bonılla. “DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT”. Maltepe Journal of Mathematics, vol. 4, no. 2, 2022, pp. 38-43, doi:10.47087/mjm.1160600.
Vancouver Bulnes J, López-bonılla JL. DIRAC’S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT. Maltepe Journal of Mathematics. 2022;4(2):38-43.

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