Using two types of computer algebra systems to solve maxwell optics problems

Abstract

To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately supported by computer algebra systems. In this paper, we distinguish between two stages of working with a mathematical model: model development and model usage. Each of these stages implies its own computer algebra system. As a model problem, we consider the problem of geometrization of Maxwell’s equations. Two computer algebra systems—Cadabra and FORM—are selected for use at different stages of investigation.

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References

  1. 1.

    Hadamard, J.S., Psychology of Invention in the Mathematical Field, Dover, 1954, 2nd ed.

    Google Scholar 

  2. 2.

    Brooks, F.P.J., No silver bullet-essence and accidents of software engineering, Proc. IFIP Tenth World Computing Conference, 1986, pp. 1069–1076.

    Google Scholar 

  3. 3.

    Penrose, R. and Rindler, W., Spinors and Space-Time: Two-Spinor Calculus and Relativistic Fields, Cambridge: Cambridge Univ. Press, 1987, vol. 1.

    Google Scholar 

  4. 4.

    Korol’kova, A.V., Kulyabov, D.S., and Sevast’yanov, L.A., Tensor computations in computer algebra systems, Program. Comput. Software, 2013, vol. 39, no. 3. pp. 135–142.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Sevastianov, L.A., Kulyabov, D.S., and Kokotchikova, M.G., An application of computer algebra system Cadabra to scientific problems of physics, Phys. Part. Nucl. Lett., 2009, vol. 6, no. 7. pp. 530–534.

    Article  Google Scholar 

  6. 6.

    Peeters, K., Cadabra: A field-theory motivated symbolic computer algebra system, Comput. Phys. Commun., 2007, vol. 176, no. 8. pp. 550–558.

    Article  MATH  Google Scholar 

  7. 7.

    Peeters, K., Introducing Cadabra: A symbolic computer algebra system for field theory problems. http://arxivorg/abs/hep-th/0701238.

  8. 8.

    Peeters, K., Symbolic field theory with Cadabra, Computeralgebra- Rundbrief, 2007, no. 41, pp. 16–19.

    MATH  Google Scholar 

  9. 9.

    Brewin, L., A brief introduction to Cadabra: A tool for tensor computations in general relativity, Comput. Phys. Commun., 2010, vol. 181, no. 3. pp. 489–498.

    Article  MATH  Google Scholar 

  10. 10.

    Tung, M.M., FORM matters: Fast symbolic computation under UNIX, Comput. Math. Appl., 2005, vol. 49, pp. 1127–1137.

    MathSciNet  Article  Google Scholar 

  11. 11.

    Vermaseren, J.A.M., Kuipers, J., Tentyukov, M., et al., FORM version 4.1 Reference Material, 2013.

    Google Scholar 

  12. 12.

    Heck, A.J.P. and Vermaseren, J.A.M., FORM for Pedestrians, Amsterdam, 2000.

    Google Scholar 

  13. 13.

    Fliegner, D., Retey, A., and Vermaseren, J.A.M., Parallelizing the symbolic manipulation program FORM. http://arxivorg/abs/hep-ph/9906426.

  14. 14.

    Tentyukov, M. and Vermaseren, J.A.M., Extension of the functionality of the symbolic program FORM by external software, Comput. Phys. Commun., 2007, vol. 176, no. 6. pp. 385–405.

    Article  Google Scholar 

  15. 15.

    Boos, E.E. and Dubinin, M.N., Problems of automatic computations for physics on colliders, Usp. Fiz. Nauk, 2010, vol. 180, no. 10. pp. 1081–1094

    Article  Google Scholar 

  16. 16.

    Bunichev, V., Kryukov, A., and Vologdin, A., Using FORM for symbolic evaluation of Feynman diagrams in CompHEP package, Nucl. Instrum. Methods Phys. Res., Sect. A, 2003, vol. 502, pp. 564–566.

    Article  Google Scholar 

  17. 17.

    Hahn, T., Generating and calculating one-loop Feynman diagrams with FeynArts, FormCalc, and Loop- Tools. http://arxivorg/abs/hep-ph/9905354.

  18. 18.

    Hahn, T., Automatic loop calculations with FeynArts, FormCalc, and LoopTools. http://arxivorg/abs/hepph/0005029.

  19. 19.

    Hahn, T. and Lang, P., FeynEdit: A tool for drawing Feynman diagrams. http://arxivorg/abs/0711.1345.

  20. 20.

    Wheeler, J.A., Neutrinos, Gravitation, and Geometry, Bologna, 1960.

    Google Scholar 

  21. 21.

    Tamm, I.E., Electrodynamics of an anisotropic medium in the special relativity theory, Zh. Russ. Fiz.- Khim. O-va., Chast Fiz., 1924, vol. 56, nos. 2–3, pp. 248–262.

    Google Scholar 

  22. 22.

    Tamm, I.E., Crystal optics of the relativity theory in connection with the geometry of a biquadratic form, Zh. Russ. Fiz.-Khim. O-va., Chast Fiz., 1925, vol. 57, nos. 3–4, pp. 209–240.

    Google Scholar 

  23. 23.

    Tamm, I.E. and Mandelstam, L.I., Elektrodynamik der anisotropen Medien in der speziellen Relativitatstheorie, Mathematische Annalen, 1925, vol. 95, no. 1. pp. 154–160.

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Plebanski, J., Electromagnetic waves in gravitational fields, Phys. Rev., 1960, vol. 118, no. 5. pp. 1396–1408.

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Felice, F., On the gravitational field acting as an optical medium, Gen. Relativ. Gravitation, 1971, vol. 2, no. 4. pp. 347–357.

    Article  Google Scholar 

  26. 26.

    Leonhardt, U., Philbin, T.G., and Haugh, N., General Relativity in Electrical Engineering, 2008, pp. 1–19.

    Google Scholar 

  27. 27.

    Leonhardt, U. and Philbin, T.G., Transformation optics and the geometry of light, Prog. Opt., 2009, vol. 53, pp. 69–152.

    Article  Google Scholar 

  28. 28.

    Kulyabov, D.S., Korolkova, A.V., and Korolkov, V.I., Maxwell’s equations in arbitrary coordinate system, Bulletin of Peoples’ Friendship University of Russia, Series “Mathematics. Information Sciences. Physics,” 2012, no. 1, pp. 96–106.

    Google Scholar 

  29. 29.

    Kulyabov, D.S., Geometrization of electromagnetic waves, Proc. Int. Conf. Mathematical Modeling and Computational Physics (MMCP), Dubna, 2013, p. 120.

    Google Scholar 

  30. 30.

    Kulyabov, D.S. and Nemchaninova, N.A., Maxwell’s equations in curvilinear coordinates, Bulletin of Peoples’ Friendship University of Russia, Series “Mathematics. Information Sciences. Physics,” 2011, no. 2, pp. 172–179.

    Google Scholar 

  31. 31.

    Minkowski, H., Die grundlagen fur die electromagnetischen vorgange in bewegten korpern, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl., 1908, no. 68, pp. 53–111.

    MATH  Google Scholar 

  32. 32.

    Stratton, J.A., Electromagnetic Theory, Wiley, 2007.

    Google Scholar 

  33. 33.

    Fulton W., Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge: Cambridge Univ. Press, 1997.

    Google Scholar 

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Correspondence to D. S. Kulyabov.

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Original Russian Text © D.S. Kulyabov, 2016, published in Programmirovanie, 2016, Vol. 42, No. 2.

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Kulyabov, D.S. Using two types of computer algebra systems to solve maxwell optics problems. Program Comput Soft 42, 77–83 (2016). https://doi.org/10.1134/S0361768816020043

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Keywords

  • Form System
  • Young Diagram
  • Message Passing Interface
  • Computer Algebra System
  • Vector Analysis