Maxwell Geometric Dynamics
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.
KeywordsNewton Method Finite Difference Scheme PDEs System Linear PDEs Euclidean Scalar Product
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.
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