Advertisement

Symmetries and Noether’s Theorem in MHD

  • Gary Webb
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 946)

Abstract

In this chapter we discuss Noether’s first theorem in MHD. The analysis is similar to that in Padhye (1998) and Webb et al. (2005b) We consider the Lagrangian form of the action ( 10.11), namely

References

  1. Akhatov, I., Gazizov, R., Ibragimov, N.: Nonlocal Symmetries, Heuristic Approach (English Translation). J. Sov. Math. 55(1), 1401 (1991)Google Scholar
  2. Anco, S.C., Bluman, G.W.: Direct Construction Method for Conservation Laws of Partial Differential Equations. Part I: Examples of Conservation Law Classification. Eur. J. Appl. Math. 13, 545–566 (2002a)Google Scholar
  3. Anco, S.C., Bluman, G.W.: Direct Construction Method for Conservation Laws of Partial Differential Equations. Part II: General Treatment. Eur. J. Appl. Math. 13, 567–585 (2002b)Google Scholar
  4. Bluman, G.: Nonlocal Extensions of Similarity Methods. J. Nonlinear Math. Phys. 15, 1–24 (2008)Google Scholar
  5. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Applied Mathematical Sciences 81. Springer, New York (1989)Google Scholar
  6. Cheviakov, A.F., Anco, S.C.: Analytical Properties and Exact Solutions of Static Plasma Equilibrium Systems in Three Dimensions. Phys. Lett. A 372, 1363–1373 (2008)Google Scholar
  7. Dewar, R.L.: Interaction Between Hydromagnetic Waves and a Time Dependent Inhomogeneous Medium. Phys. Fluids 13(11), 2710–2720 (1970)Google Scholar
  8. Fuchs, J.C.: Symmetry Groups of Similarity Solutions of the MHD Equations. J. Math. Phys. 32, 1703–1708 (1991)Google Scholar
  9. Golovin, S.V.: Natural Curvilinear Coordinates for Ideal MHD Equations. Non-stationary Flows with Constant Pressure. Phys. Lett. A c375, 283–290 (2011)Google Scholar
  10. Grundland, A.M., Lalague, L.: Lie Subgroups of Fluid Dynamics and Magnetohydrodynamics Equations. Can. J. Phys. 73, 463–477 (1995)Google Scholar
  11. Henyey, F.S.: Canonical Construction of a Hamiltonian for Dissipation-Free Magnetohydrodynamics. Phys. Rev. A 26(1), 480–483 (1982)Google Scholar
  12. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)Google Scholar
  13. Ibragimov, N.H., Kara, A.H., Mahomed, F.M.: Lie-Bäcklund and Noether Symmetries with Applications. Nonlinear Dyn. 15, 115–136 (1998)Google Scholar
  14. Morrison, P.J.: Poisson Brackets for Fluids and Plasmas. In: Tabor, M., Treve, Y.M. (eds.) Mathematical Methods in Hydrodynamics and Integrability of Dynamical Systems. AIP Conference Proceedings 88, pp. 13–46. American Institute of Physics (1982)Google Scholar
  15. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)Google Scholar
  16. Olver, P.J., Nutku, Y.: Hamiltonian Structures for Systems of Hyperbolic Conservation Laws. J. Math. Phys. 29, 1610–1619 (1988)Google Scholar
  17. Padhye, N.S.: Topics in lagrangian and hamiltonian fluid dynamics: relabeling symmetry and ion acoustic wave stability. Ph.D. Dissertation, University of Texas at Austin (1998)Google Scholar
  18. Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)Google Scholar
  19. Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)ADSGoogle Scholar
  20. Sjöberg, A., Mahomed, F.M.: Nonlocal Symmetries and Conservation Laws for One Dimensional Gas Dynamics Equations. Appl. Math. Comput. 150, 379–397 (2004)MathSciNetzbMATHGoogle Scholar
  21. Webb, G.M., Zank, G.P.: Fluid Relabelling Symmetries, Lie Point Symmetries and the Lagrangian Map in Magnetohydrodynamics and Gas Dynamics. J. Phys. A. Math. Theor. 40, 545–579 (2007). https://doi.org/10.1088/1751-8113/40/3/013 zbMATHGoogle Scholar
  22. Webb, G.M., Zank, G.P.: Scaling Symmetries, Conservation Laws and Action Principles in One-Dimensional Gas Dynamics. J. Phys. A. Math. Theor. 42, 475205 (23 pp.) (2009)Google Scholar
  23. Webb, G.M., Zank, G.P., Kaghashvili, E.Kh., Ratkiewicz, R.E.: Magnetohydrodynamic Waves in Non-uniform Flows I: A Variational Approach. J. Plasma Phys. 71(6), 785–809 (2005a). https://doi.org/10.1017/S00223778050003739 ADSCrossRefGoogle Scholar
  24. Webb, G.M., Zank, G.P., Kaghashvili, E.Kh., Ratkiewicz, R.E.: Magnetohydrodynamic Waves in Non-uniform Flows II: Stress Energy Tensors, Conservation Laws and Lie Symmetries. J. Plasma Phys. 71, 811–857 (2005b). https://doi.org/10.1017/S00223778050003740 Google Scholar
  25. Webb, G.M., Pogorelov, N.V., Zank, G.P.: MHD Simple Waves and the Divergence Wave. In: Twelfth International Solar Wind Conference, St. Malo. AIP Conference Proceedings 1216, pp. 300–303 (2009). https://doi.org/10.1063/1.3396300 ADSGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Gary Webb
    • 1
  1. 1.CSPARThe University of Alabama in HuntsvilleHuntsvilleUSA

Personalised recommendations