Skip to main content

Part of the book series: Lecture Notes in Physics ((LNP,volume 946))

  • 1480 Accesses

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Akhatov, I., Gazizov, R., Ibragimov, N.: Nonlocal Symmetries, Heuristic Approach (English Translation). J. Sov. Math. 55(1), 1401 (1991)

    Google Scholar 

  • 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 

  • 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 

  • Bluman, G.: Nonlocal Extensions of Similarity Methods. J. Nonlinear Math. Phys. 15, 1–24 (2008)

    Google Scholar 

  • Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Applied Mathematical Sciences 81. Springer, New York (1989)

    Google Scholar 

  • 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 

  • Dewar, R.L.: Interaction Between Hydromagnetic Waves and a Time Dependent Inhomogeneous Medium. Phys. Fluids 13(11), 2710–2720 (1970)

    Google Scholar 

  • Fuchs, J.C.: Symmetry Groups of Similarity Solutions of the MHD Equations. J. Math. Phys. 32, 1703–1708 (1991)

    Google Scholar 

  • 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 

  • Grundland, A.M., Lalague, L.: Lie Subgroups of Fluid Dynamics and Magnetohydrodynamics Equations. Can. J. Phys. 73, 463–477 (1995)

    Google Scholar 

  • Henyey, F.S.: Canonical Construction of a Hamiltonian for Dissipation-Free Magnetohydrodynamics. Phys. Rev. A 26(1), 480–483 (1982)

    Google Scholar 

  • Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)

    Google Scholar 

  • Ibragimov, N.H., Kara, A.H., Mahomed, F.M.: Lie-Bäcklund and Noether Symmetries with Applications. Nonlinear Dyn. 15, 115–136 (1998)

    Google Scholar 

  • 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 

  • Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)

    Google Scholar 

  • Olver, P.J., Nutku, Y.: Hamiltonian Structures for Systems of Hyperbolic Conservation Laws. J. Math. Phys. 29, 1610–1619 (1988)

    Google Scholar 

  • 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 

  • Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)

    Google Scholar 

  • Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)

    ADS  Google Scholar 

  • Sjöberg, A., Mahomed, F.M.: Nonlocal Symmetries and Conservation Laws for One Dimensional Gas Dynamics Equations. Appl. Math. Comput. 150, 379–397 (2004)

    MathSciNet  MATH  Google Scholar 

  • 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

    MATH  Google Scholar 

  • 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 

  • 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

    Article  ADS  Google Scholar 

  • 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 

  • 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

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Webb, G. (2018). Symmetries and Noether’s Theorem in MHD. In: Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws. Lecture Notes in Physics, vol 946. Springer, Cham. https://doi.org/10.1007/978-3-319-72511-6_11

Download citation

Publish with us

Policies and ethics