- PII
- 10.31857/S0132347424020078-1
- DOI
- 10.31857/S0132347424020078
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume / Issue number 2
- Pages
- 66-73
- Abstract
- Different implementations of Clifford algebra: spinors, quaternions, and geometric algebra, are used to describe physical and technical systems. The geometric algebra formalism is a relatively new approach, destined to be used primarily by engineers and applied researchers. In a number of works, the authors examined the implementation of the geometric algebra formalism for computer algebra systems. In this article, the authors extend elliptic geometric algebra to hyperbolic space-time algebra. The results are illustrated by different representations of Maxwell’s equations. Using a computer algebra system, Maxwell’s vacuum equations in the space-time algebra representation are converted to Maxwell’s equations in vector formalism. In addition to practical application, the authors would like to draw attention to the didactic significance of these studies.
- Keywords
- геометрическая алгебра алгебра Клиффорда система компьютерной алгебры SymPy Galgebra
- Date of publication
- 15.04.2024
- Year of publication
- 2024
- Number of purchasers
- 0
- Views
- 41
References
- 1. Геворкян М.Н., Королькова А.В., Кулябов Д.С. Реализация геометрической алгебры в системах символьных вычислений // Программирование. 2023. № 1. С. 48–55.
- 2. Геворкян М.Н., Демидова А.В., Велиева Т.Р. Аналитико-численная реализация алгебры поливекторов на языке Julia // Программирование. 2022. № 1. С. 54–64.
- 3. Велиева Т.Р., Геворкян М.Н., Демидова А.В. Аппарат геометрической алгебры и кватернионов в системах символьных вычислений для описания вращений в евклидовом пространстве // Журнал вычислительной математики и математической физики. 2023. Т. 63. № 1. С. 31–42.
- 4. Королькова А.В., Геворкян М.Н., Кулябов Д.С., Севастьянов Л.А. Средства компьютерной алгебры для геометризации уравнений Максвелла // Программирование. 2023. Т. 49. № 4. С. 33–38.
- 5. Kulyabov D. S. Using two Types of Computer Algebra Systems to Solve Maxwell Optics Problems // Programming and Computer Software. 2016. V. 42. № 2. P. 77–83. arXiv: 1605.00832.
- 6. Kulyabov D.S., Korolkova A.V., Sevastianov L.A. et al. Algorithm for Lens Calculations in the Geometrized Maxwell Theory // Saratov Fall Meeting 2017: Laser Physics and Photonics XVIII; and Computational Biophysics and Analysis of Biomedical Data IV / Ed. by Vladimir L. Derbov, Dmitry E. Postnov. Vol. 10717 of Progress in Biomedical Optics and Imaging – Proceedings of SPIE. Saratov: SPIE, 2018. 4. P. 107170Y.1–6. arXiv : 1806.01643.
- 7. Grassmann H.G. Die Mechanik nach den Principien der Ausdehnungslehre // Mathematische Annalen. 1877. 6. Bd. 12, H. 2. S. 222–240.
- 8. Kuipers J.B. Quaternions And Rotation Sequences. Princeton, New Jersey: Princeton University Press, 2002. 400 p.
- 9. Clifford W.K. Applications of Grassmann’s Extensive Algebra // American Journal of Mathematics. 1878. V. 1. № 4. P. 350–358.
- 10. GAlgebra — Symbolic Geometric Algebra/ Calculus package for SymPy. 2023. https://galgebra.readthedocs.io/en/latest/index.html.
- 11. Velieva T.R., Gevorkyan M.N., Demidova A.V. et al. Geometric Algebra and Quaternion Techniques in Computer Algebra Systems for Describing Rotations in Eucledean Space // Computational Mathematics and Mathematical Physics. 2023. V. 63. № 1. P. 29–39.
- 12. Sandon D. Symbolic Computation with Python and SymPy. 2021. V. 1. 580 p.
- 13. Sandon D. Symbolic Computation with Python and SymPy. 2021. V. 2. 429 p.
- 14. The international system of units (SI) / Ed. by David B. Newell, Eite Tiesinga. NIST Special Publication 330. National Institute of Standards and Technology, 2019. Aug. 122 p.
- 15. Dorst L., Fontijne D., Mann S. Geometric algebra for computer science (with errata). The Morgan Kaufmann Series in Computer Graphics. 1 edition. Morgan Kaufmann, 2007.
- 16. de Sabbata V., Datta B.K. Geometric Algebra and Applications to Physics. Taylor & Francis, 2006. 12. 184 p.
- 17. Rosn A. Geometric Multivector Analysis. Springer International Publishing, 2019. 465 p.
- 18. Rodrigues Jr W.A., de Oliveira E.C. The Many Faces of Maxwell, Dirac and Einstein Equations. Springer International Publishing, 2016. V. 922 of Lecture Notes in Physics. 587 p.
- 19. Doran C., Lasenby A. Geometric Algebra for Physicists. Cambridge University Press, 2003. May. 578 p.
- 20. Chisolm E. Geometric Algebra. 2012. arXiv : 1205.5935.
- 21. Lasenby A., Doran C., Arcaute E. Applications of Geometric Algebra in Electromagnetism, Quantum Theory and Gravity // Clifford Algebras / Ed. by R. Abamowicz. Birkhuser Boston, 2004. Vol. 34 of Progress in Mathematical Physics. 467–489 p.
- 22. Toomey D. Learning Jupyter. Packt Publishing Ltd., 2016. 305 p.
