Stokes Representation for the Solutions of Maxwell-Vlasov

Monica Pîrvan, Constantin Udrişte

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DOI: 10.4236/jemaa.2011.31002                

Abstract

Maxwell-Vlasov PDEs system describes the dynamics of plasma consisting of charged particles with long-range inter-action. Their solutions can be written using some Stokes potentials. Section 1 presents the experimental devices which can produce a magnetic trap. Magnetic geometric dynamic provides mathematical tools for describing the magnetic flow (see [1-7]). Stokes representation for the solutions of PDEs as Maxwell PDEs or Maxwell-Vlasov PDEs are used analyzing electromagnetic energy in magnetic traps. Section 2 studies Maxwell-Vlasov PDEs system. Stokes represen-tation of its solutions, using Maximum Principle for a multitime optimal control problem, is obtained. Section 3 dis-cusses a method for changing a given ODEs system into a geodesic motion under a gyroscopic field of forces (geomet-ric dynamics). Section 4 proposes a modified form for Maxwell-Vlasov PDEs, by replacing the classical gyroscopic force with the one appearing in geometric dynamics. Stokes representation for the solutions of modified Max-well-Vlasov PDEs is also obtained.

Keywords

Maxwell-Vlasov PDEs, Magnetic Traps, Dynamics of Plasma, Gyroscopic Field of Forces, Dynamic Systems

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. Udri?te, “Geometric Dynamics,” Kluwer Academic Publishers, Article Southeast Asian Bulletin of Mathematics, Springer-Verlag, New York, Vol. 24, 2000, pp. 313-322.
[2]	C. Udri?te, “Nonclassical Lagrangian Dynamics and Potential Maps,” Proceedings of the Conference on Mathematics in Honour of Profesor Radu Ro?ca at the Ocassion of His Ninetieth Birthday, Belgium, 11-16 December 1999.
[3]	C. Udri?te and M. Postolache, “Atlas of Magnetic Geometric Dynamics,” Geometry Balkan Press, Bucharest, 2001.
[4]	C. Udri?te, “Tools of Geometric Dynamics,” Buletinul Institutului de Geodinamic?, Academia Romana, Vol. 14, No. 4, 2003, pp. 1-26.
[5]	C. Udri?te, “Maxwell Geometric Dynamics,” European Computing Conference, Athens, Greece, 24-26 September 2007, pp. 1597-1609.
[6]	C. Udri?te and M. P?rvan, “Magnetic Dynamics around a Configuration of Two Square AntiHelmholtz Coils,” Proceedings of 7th WSEAS International Conference Nonlinear Analysis, Non-linear Systems and Chaos (NOLASC 08), Corfu, Greece, 26-28 October 2008, pp. 222-227.
[7]	C. Udri?te and M. P?rvan, “Magnetic geometric dynamics around Tornado Trap,” Proceedings of the 10th WSEAS International Conference Mathematical Computational Mathematical Science and Engineering (MACMESE 08), Bucharest, 7-9 November 2008, pp. 312-317.
[8]	P. Preuss, “Angels, Demons, and Antihydrogen: The Real Science of Anti-Atoms,” 2009. http://www.symmetry magzine.org/breaking/2009/05/07/angels-demons-and-antihydrogen-the-rel-science-of-anti-atoms/
[9]	C. Udri?te and L. Matei, “Teorii Lagrange-Hamilton,” (in Romanian) Monographs and Textbooks 8, Geometry Balkan Press, Bucharest, 2008.
[10]	C. Udri?te, “Simplified Multitime Maximum Principle,” Balkan Journal of Geometry and Its Applications, Vol. 14, No. 1, 2009, pp. 102-119.
[11]	C. Udri?te, M. Ferrara and D. Opri?, “Economic geometric dynamics,” Monographs and Textbooks 6, Geometry Balkan Press, Bucharest, 2004.
[12]	C. Udri?te, “Geodesic motion in a gyroscopic field of forces,” Tensor, N. S., Vol. 66, No. 3, 2005, pp. 215-228.
[13]	H. Cendra, D. D. Holm, M. J. W. Hoyle and J. E. Marsden, “The Maxwell-Vlasov Equations in Euler-Poincaré form,” Journal of Mathematical Physics, Vol. 39, No. 6, 1998, pp. 3138-3157. doi:10.1063/1.532244
[14]	V. Ciancio and C. Udri?te, “Ioffe-?tef?nescu Magnetic Trap,” Revue Roumaine de Sciences Techniques-Serie Electrotechnique et Energetique, Vol. 49, No. 2, 2004, pp. 157-176.
[15]	Y. Guo, “Global Weak Solutions of the Vlasov-Maxwell System with Boundary Conditions,” Communications in Mathematical Physics, Vol. 154, No. 2, 1993, pp. 245- 263. doi:10.1007/BF02096997
[16]	E. Marsden and T. S. Ra?iu, “Intro-Duction to Mechanics and Symmetry,” 2nd Edition, Springer, New York, 1999.
[17]	C. Neagu and C. Udri?te, “From PDE Systems and Metrics to Geometric Multi-time Field Theories,” Seminarul de Mecanic?, Sisteme Dinamice Diferen?iale, 79, Universitatea de Vest din Timi?oara, 2001.
[18]	M. P?rvan and C.Udri?te, “Optimal Control of Electromagnetic Energy,” Balkan Journal of Geometry and Its Applications, Vol. 15, No. 1, 2010, pp. 131-141.
[19]	C. Udri?te, “Nonholonomic Aapproach of Multitime Maximum Principle,” Balkan Journal of Geometry and Its Applications, Vol. 14, No. 2, 2009, pp. 101-116.
[20]	D. Wang and X. Hu, “Weak Stability and Hydrodynamic Limit of Vlasov-Maxwell-Bolzmann Equations,” SIAM Conference on Analysis of Partial Differential Equations, Miami, Florida, 7-10 December 2009, p. 78.