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.
References
Hadamard, J.S., Psychology of Invention in the Mathematical Field, Dover, 1954, 2nd ed.
Brooks, F.P.J., No silver bullet-essence and accidents of software engineering, Proc. IFIP Tenth World Computing Conference, 1986, pp. 1069–1076.
Penrose, R. and Rindler, W., Spinors and Space-Time: Two-Spinor Calculus and Relativistic Fields, Cambridge: Cambridge Univ. Press, 1987, vol. 1.
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.
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.
Peeters, K., Cadabra: A field-theory motivated symbolic computer algebra system, Comput. Phys. Commun., 2007, vol. 176, no. 8. pp. 550–558.
Peeters, K., Introducing Cadabra: A symbolic computer algebra system for field theory problems. http://arxivorg/abs/hep-th/0701238.
Peeters, K., Symbolic field theory with Cadabra, Computeralgebra- Rundbrief, 2007, no. 41, pp. 16–19.
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.
Tung, M.M., FORM matters: Fast symbolic computation under UNIX, Comput. Math. Appl., 2005, vol. 49, pp. 1127–1137.
Vermaseren, J.A.M., Kuipers, J., Tentyukov, M., et al., FORM version 4.1 Reference Material, 2013.
Heck, A.J.P. and Vermaseren, J.A.M., FORM for Pedestrians, Amsterdam, 2000.
Fliegner, D., Retey, A., and Vermaseren, J.A.M., Parallelizing the symbolic manipulation program FORM. http://arxivorg/abs/hep-ph/9906426.
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.
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
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.
Hahn, T., Generating and calculating one-loop Feynman diagrams with FeynArts, FormCalc, and Loop- Tools. http://arxivorg/abs/hep-ph/9905354.
Hahn, T., Automatic loop calculations with FeynArts, FormCalc, and LoopTools. http://arxivorg/abs/hepph/0005029.
Hahn, T. and Lang, P., FeynEdit: A tool for drawing Feynman diagrams. http://arxivorg/abs/0711.1345.
Wheeler, J.A., Neutrinos, Gravitation, and Geometry, Bologna, 1960.
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.
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.
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.
Plebanski, J., Electromagnetic waves in gravitational fields, Phys. Rev., 1960, vol. 118, no. 5. pp. 1396–1408.
Felice, F., On the gravitational field acting as an optical medium, Gen. Relativ. Gravitation, 1971, vol. 2, no. 4. pp. 347–357.
Leonhardt, U., Philbin, T.G., and Haugh, N., General Relativity in Electrical Engineering, 2008, pp. 1–19.
Leonhardt, U. and Philbin, T.G., Transformation optics and the geometry of light, Prog. Opt., 2009, vol. 53, pp. 69–152.
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.
Kulyabov, D.S., Geometrization of electromagnetic waves, Proc. Int. Conf. Mathematical Modeling and Computational Physics (MMCP), Dubna, 2013, p. 120.
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.
Minkowski, H., Die grundlagen fur die electromagnetischen vorgange in bewegten korpern, Nachr. Ges. Wiss. Goettingen, Math.-Phys. Kl., 1908, no. 68, pp. 53–111.
Stratton, J.A., Electromagnetic Theory, Wiley, 2007.
Fulton W., Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge: Cambridge Univ. Press, 1997.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D.S. Kulyabov, 2016, published in Programmirovanie, 2016, Vol. 42, No. 2.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768816020043