Abstract
This paper describes the least squares approximations of the solutions of Maxwell PDEs via their Euler-Lagrange prolongations. Section 5.1 recalls the variational problem in electrodynamics. Section 69.2 describes the Ibragimov-Maxwell Lagrangian. Section 69.3 gives the Ibragimov-Udrişte-Maxwell Lagrangian and finds its Euler-Lagrange PDEs system. Section 5.4 finds the Euler-Lagrange PDEs system associated to Udrişte-Maxwell Lagrangian, and shows that the waves are particular solutions. Section 69.5 studies the discrete Maxwell geometric dynamics. Section 69.6 performs Von Neumann analysis of the associated difference scheme. Section 69.7 addresses an open problem regarding our theory in the context of differential forms on a Riemannian manifold. Section 69.8 underlines the importance of the least squares Lagrangian.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bilbao S (2004) Wave and scattering methods for numerical simulation. John Wiley and Sons, Chicester, UK, p 368
Ibragimov NH (2006) Integrating factors, adjoint equations and Lagrangians. J Math Anal Appl 318(2):742–757
Kempner AJ (1935) On the complex roots of algebraic equations. Bull Amer Math Soc 41(12):809–843
Udrişte C (1999) Nonclassical Lagrangian dynamics and potential maps. In: Conference in Mathematics in Honour of Professor Radu Roşca on the occasion of his Ninetieth Birthday, Katholieke University Brussel, Katholieke University Leuwen, Belgium, http://xxx.lanl.gov/math.DS/0007060 (2000)
Udrişte C (2000) Solutions of ODEs and PDEs as potential maps using first order Lagrangians. Centenial Vrânceanu, Romanian Academy, University of Bucharest, http://xxx.lanl.gov/math.DS/0007061 (2000); Balkan J Geometry Appl (2001) 6(1):93–108
Udrişte C (2003) Tools of geometric dynamics. Buletinul Institutului de Geodinamică, Academia Română, 14(4):1–26; In: Proceedings of the XVIII Workshop on Hadronic Mechanics, honoring the 70-th birthday of Prof. R. M. Santilli, the originator of hadronic mechanics, University of Karlstad, Sweden, June 20–22, 2005; Valer Dvoeglazov, Tepper L. Gill, Peter Rowland, Erick Trell, Horst E. Wilhelm (Eds) Hadronic Press, International Academic Publishers, December 2006, ISBN 1-57485-059-28, pp 1001–1041
Udrişte C (2003-2004) From integral manifolds and metrics to potential maps. Atti dell'Academia Peloritana dei Pericolanti, Classe I di Scienze Fis. Mat et Nat 81–82:1–16, C1A0401008
Udrişte C (2005) Geodesic motion in a gyroscopic field of forces. Tensor, N. S. 66(3):215–228
Udrişte C (2006) Multi-time maximum principle. Short communication at International Congress of Mathematicians, Madrid
Udrişte C, Ferrara M, Opriş D (2004) Economic geometric dynamics. Monographs and Textbooks 6, Geometry Balkan Press
Udrişte C, Postolache M (2000) Least squares problem and geometric dynamics. Italian J Pure Appl Math 15:77–88
Udrişte C, Postolache M, Ţevy I (2001) Integrator for Lagrangian dynamics. Balkan J Geometry Appl 6(2):109–115
Udrişte C, Ţevy I (2007) Multi-time Euler-Lagrange-Hamilton theory. WSEAS Trans Math 6(6):701–709
Udrişte C (2007) Multi-time controllability, observability and bang-bang principle. In: 6th Congress of Romanian Mathematicians, Bucharest, Romania
Udrişte C, Ţevy I (2007) Multi-Time Euler-Lagrange Dynamics. In: Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation (ISTASC'07), Athens, Greece, pp 66–71
Acknowledgements
Partially supported by Grant CNCSIS 86/ 2007 and by 15th Italian-Romanian Executive Programme of S&T Co-operation for 2006-2008, University Politehnica of Bucharest.
We are very appreciative to Prof. Dr. Nikos Mastorakis, Prof. Dr. Valeriu Prepeliţă and Lecturer Romeo Bercia for their suggestions regarding the improvement of numerical aspects for the least squares approximations via Von Neumann analysis.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media, LLC
About this paper
Cite this paper
Udrişte, C. (2009). Maxwell Geometric Dynamics. In: Mastorakis, N., Mladenov, V., Kontargyri, V. (eds) Proceedings of the European Computing Conference. Lecture Notes in Electrical Engineering, vol 28. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-85437-3_69
Download citation
DOI: https://doi.org/10.1007/978-0-387-85437-3_69
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-84818-1
Online ISBN: 978-0-387-85437-3
eBook Packages: EngineeringEngineering (R0)