Using two types of computer algebra systems to solve Maxwell optics problems. (English. Russian original) Zbl 1344.78021
Program. Comput. Softw. 42, No. 2, 77-83 (2016); translation from Programmirovanie 42, No. 2 (2016).
Summary: 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.
MSC:
| 78M25 | Numerical methods in optics (MSC2010) |
| 68W30 | Symbolic computation and algebraic computation |
| 78A50 | Antennas, waveguides in optics and electromagnetic theory |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
Keywords:
computer algebra systems; Maxwell optics problems; geometrization of Maxwell’s equation; waveguide; inverse problemReferences:
| [1] | Hadamard, J.S., Psychology of Invention in the Mathematical Field, Dover, 1954, 2nd ed. · Zbl 0056.00101 |
| [2] | Brooks, F. P.J., No silver bullet-essence and accidents of software engineering, 1069-1076 (1986) |
| [3] | Penrose, R. and Rindler, W., Spinors and Space-Time: Two-Spinor Calculus and Relativistic Fields, Cambridge: Cambridge Univ. Press, 1987, vol. 1. · Zbl 0663.53013 |
| [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. · Zbl 1301.68280 · doi:10.1134/S0361768813030031 |
| [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. · doi:10.1134/S1547477109070073 |
| [6] | Peeters, K., Cadabra: A field-theory motivated symbolic computer algebra system, Comput. Phys. Commun., 2007, vol. 176, no. 8. pp. 550-558. · Zbl 1196.68333 · doi:10.1016/j.cpc.2007.01.003 |
| [7] | Peeters, K., Introducing Cadabra: A symbolic computer algebra system for field theory problems. http://arxivorg/abs/hep-th/0701238. · Zbl 1196.68333 |
| [8] | Peeters, K., Symbolic field theory with Cadabra, Computeralgebra- Rundbrief, 2007, no. 41, pp. 16-19. · Zbl 1196.68333 |
| [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. · Zbl 1206.83007 · doi:10.1016/j.cpc.2009.10.020 |
| [10] | Tung, M.M., FORM matters: Fast symbolic computation under UNIX, Comput. Math. Appl., 2005, vol. 49, pp. 1127-1137. · doi:10.1016/j.camwa.2004.07.023 |
| [11] | Vermaseren, J.A.M., Kuipers, J., Tentyukov, M., et al., FORM version 4.1 Reference Material, 2013. |
| [12] | Heck, A.J.P. and Vermaseren, J.A.M., FORM for Pedestrians, Amsterdam, 2000. |
| [13] | Fliegner, D., Retey, A., and Vermaseren, J.A.M., Parallelizing the symbolic manipulation program FORM. http://arxivorg/abs/hep-ph/9906426. · Zbl 1196.68333 |
| [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. · doi:10.1016/j.cpc.2006.11.007 |
| [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 · doi:10.3367/UFNr.0180.201010d.1081 |
| [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. · doi:10.1016/S0168-9002(03)00503-5 |
| [17] | Hahn, T., Generating and calculating one-loop Feynman diagrams with FeynArts, FormCalc, and Loop- Tools. http://arxivorg/abs/hep-ph/9905354. |
| [18] | Hahn, T., Automatic loop calculations with FeynArts, FormCalc, and LoopTools. http://arxivorg/abs/hepph/0005029. |
| [19] | Hahn, T. and Lang, P., FeynEdit: A tool for drawing Feynman diagrams. http://arxivorg/abs/0711.1345. |
| [20] | Wheeler, J.A., Neutrinos, Gravitation, and Geometry, Bologna, 1960. |
| [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. |
| [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. |
| [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. · JFM 51.0716.04 |
| [24] | Plebanski, J., Electromagnetic waves in gravitational fields, Phys. Rev., 1960, vol. 118, no. 5. pp. 1396-1408. · Zbl 0092.45102 · doi:10.1103/PhysRev.118.1396 |
| [25] | Felice, F., On the gravitational field acting as an optical medium, Gen. Relativ. Gravitation, 1971, vol. 2, no. 4. pp. 347-357. · doi:10.1007/BF00758153 |
| [26] | Leonhardt, U., Philbin, T.G., and Haugh, N., General Relativity in Electrical Engineering, 2008, pp. 1-19. |
| [27] | Leonhardt, U. and Philbin, T.G., Transformation optics and the geometry of light, Prog. Opt., 2009, vol. 53, pp. 69-152. · doi:10.1016/S0079-6638(08)00202-3 |
| [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. |
| [29] | Kulyabov, D. S., Geometrization of electromagnetic waves, 120 (2013) |
| [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. |
| [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. · JFM 39.0909.02 |
| [32] | Stratton, J.A., Electromagnetic Theory, Wiley, 2007. · JFM 67.1119.01 |
| [33] | Fulton W., Young Tableaux: With Applications to Representation Theory and Geometry, Cambridge: Cambridge Univ. Press, 1997. · Zbl 0878.14034 |
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