- PII
- 10.31857/S0132347423020127-1
- DOI
- 10.31857/S0132347423020127
- Publication type
- Status
- Published
- Authors
- Volume/ Edition
- Volume / Issue number 4
- Pages
- 33-38
- Abstract
- Calculations of optical devices in the geometrized Maxwell’s theory use well-known formalisms of general theory of relativity and differential geometry. In particular, for such calculations it is required to know the analytical form of the geodesic equations, which leads to the need to calculate a large number of monotonous mathematical expressions. One of the purposes of computer algebra is to facilitate the researcher’s work by automating cumbersome symbolic computations. Thus, the use of computer algebra systems seems to be quite an obvious way. Several free implementations of symbolic computations for the apparatus of general relativity are considered. A practical example of symbolic computations for the geometrized Maxwell’s theory is given.
- Keywords
- Date of publication
- 01.07.2023
- Year of publication
- 2023
- Number of purchasers
- 0
- Views
- 46
References
- 1. Тамм И.Е. Электродинамика анизотропной среды в специальной теории относительности // Журнал Русского физико-химического общества. Часть физическая. 1924. Т. 56. № 2–3. С. 248–262.
- 2. Тамм И.Е. Кристаллооптика теории относительности в связи с геометрией биквадратичной формы // Журнал Русского физикохимического общества. Часть физическая. 1925. Т. 57. № 3–4. С. 209–240.
- 3. Mandelstam L.I., Tamm I.Y. Elektrodynamik der anisotropen medien in der speziellen relativittstheorie // Mathematische Annalen. 1925. Bd. 95. H. 1. S. 154–160.
- 4. Gordon W. Zur Lichtfortpflanzung nach der Relativita‥tstheorie // Annalen der Physik. 1923. Bd. 72. S. 421–456.
- 5. Plebanski J. Electromagnetic waves in gravitational fields // Physical Review. 1960. V. 118. № 5. P. 1396–1408.
- 6. Felice F. On the Gravitational Field Acting as an Optical Medium // General Relativity and Gravitation. 1971. V. 2. № 4. P. 347–357.
- 7. Smolyaninov I.I. Metamaterial ‘Multiverse’ // Journal of Optics. 2011. V. 13. № 2. P. 024004.
- 8. Pendry J.B., Schurig D., Smith D.R. Controlling Electromagnetic Fields // Science. 2006. V. 312. № 5781. P. 1780–1782.
- 9. Schurig D., Pendry J.B., Smith D.R. Calculation of Material Properties and Ray Tracing in Transformation Media // Optics express. 2006. V. 14. № 21. P. 9794–9804.
- 10. Leonhardt U. Optical Conformal Mapping // Science. 2006. V. 312. № June. P. 1777–1780.
- 11. Leonhardt U., Philbin T.G. Transformation Optics and the Geometry of Light // Progress in Optics. 2009. V. 53. P. 69–152.
- 12. Foster R., Grant P., Hao Y. et al. Spatial Transformations: from Tundamentals to Applications // Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2015. 8. V. 373. № 2049. P. 20140365.
- 13. Kulyabov D.S., Korolkova A.V., Sevastianov L.A. A naive geometrization of maxwell’s equations // The 15th small triangle meeting of Theoretical Physics. Star Lesn, 2013. P. 104–111.
- 14. Кулябов Д.С., Королькова А.В., Севастьянов Л.А. Простейшая геометризация уравнений Максвелла // Вестник РУДН. Серия. Математика. Информатика. Физика. 2014. № 2. С. 115–125.
- 15. Kulyabov D.S., Korolkova A.V., Sevastianov L.A. et al. Algorithm for lens calculations in the geometrized maxwell theory // Saratov Fall Meeting 2017. V. 10717 of Proceedings of SPIE. Saratov : SPIE, 2018. 4. P. 107170Y.1–6.
- 16. Королькова А.В., Кулябов Д.С., Севастьянов Л.А. Тензорные расчеты в системах компьютерной алгебры // Программирование. 2013. № 3. С. 47–57.
- 17. Кулябов Д.С., Королькова А.В., Севастьянов Л.А. Новые возможности второй версии пакета компьютерной алгебры cadabra // Программирование. 2019. № 2. С. 41–48.
- 18. Sandon D. Symbolic Computation with Python and SymPy. Independently published, 2021. ISBN: 979-8489815208.
- 19. Диваков Д.В., Тютюнник А.А. Символьное исследование спектральных характеристик направляемых мод плавно-нерегулярных волноводов // Программирование. 2022. № 2. С. 23–32.
- 20. Sympy. 2022. URL: http://www.sympy.org/ru/index.html.
- 21. Project jupyter. 2022. URL: https://jupyter.org/.
- 22. Einsteinpy–making einstein possible in python. 2022. URL:https://einsteinpy-einsteinpy.readthedocs.io/en /latest/index.html.
- 23. Gravipy tensor calculus package for general relativity based on sympy. 2022. URL: https://github.com/wojciechczaja/GraviPy.
- 24. Bruns H. Das Eikonal. Leipzig: S. Hirzel, 1895. Bd. 35.
- 25. Borovskikh A.V. The two-dimensional eikonal equation // Siberian Math. J. 2006. V. 47. P. 813–834.
- 26. Moskalensky E.D. Finding exact solutions to the two-dimensional eikonal equation // Num. Anal. Appl. 2009. V. 2. P. 201–209.
- 27. Kabanikhin S.I., Krivorotko O.I. Numerical solution eikonal equation // Sib. Elektron. Mat. Izv. 2013. V. 10. P. 28–34.
- 28. Kulyabov D.S., Korolkova A.V., Velieva T.R., Gevorkyan M.N. Numerical analysis of eikonal equation // Saratov Fall Meeting 2018. Vol. 11066 of Proceedings of SPIE. Saratov: SPIE, 2019. 6. P. 56.
