The Lagrangian Map

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


In this chapter we give a synopsis of Lagrangian MHD, as initially developed by Newcomb (1962). The analysis is also based on the work of Webb et al. (2005a,b), Webb and Zank (2007), and Golovin (2011) where the MHD, Lie point symmetries and the fluid relabelling symmetries were investigated using the Lagrangian map. Golovin (2011) converted the MHD equations to Lagrangian form, to obtain a vector wave equation form for the Lagrangian momentum equation, that takes into account the symmetries of the equation associated with Faraday’s equation (see also e.g. Schief (2003)).


Golovin Vector Wave Equation Lagrangian Label Mass Continuity Equation Topological Soliton Solutions 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, New York (1998)Google Scholar
  2. Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences Series 168. Springer, New York (2010)Google Scholar
  3. Bogoyavlenskij, O.I.: Symmetry Transforms for Ideal Magnetohydrodynamics Equilibria. Phys. Rev. E 66(11), 056410 (2002)ADSMathSciNetCrossRefGoogle Scholar
  4. Chandrasekhar, S.: On the Stability of the Simplest Solution of the Equations of Magnetohydrodynamics. Proc. Natl. Acad. Sci. USA 42, 273–276 (1956)ADSCrossRefzbMATHGoogle Scholar
  5. Chandrasekhar, S.: On Cosmic Magnetic Fields. Proc. Natl. Acad. Sci. USA 43, 25–27 (1957)ADSzbMATHGoogle Scholar
  6. Courant, R., Hilbert, D.: Methods of Mathematical Physics 2. Wiley, New York (1989)CrossRefzbMATHGoogle Scholar
  7. Golovin, S.V.: Analytical Description of Stationary Ideal MHD Fluid Flows with Constant Total Pressure. Phys. Lett. A 374, 901–905 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 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
  9. Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear Stability of Fluid and Plasma Equilibria. Phys. Rep. (Review section of Phys. Rev. Lett.) 123(1 and 2), 1–116 (1985).
  10. Kamchatnov, I.V.: Topological Soliton in Magnetohydrodynamics. Sov. Phys. 55(1), 69–73 (1982)MathSciNetGoogle Scholar
  11. Newcomb, W.A.: Lagrangian and Hamiltonian Methods in Magnetohydrodynamics. Nucl. Fusion Suppl. (Part 2), 451–463 (1962)Google Scholar
  12. Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)Google Scholar
  13. Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)ADSGoogle Scholar
  14. Parker, E.N.: Cosmic Magnetic Fields. Oxford University Press, New York (1979)Google Scholar
  15. Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  16. Schief, W.K.: Hidden Integrability in Ideal Magnetohydrodynamics: The Pohlmeyer-Lund-Regge Model. Phys. Plasmas 10, 2677–2685 (2003)ADSMathSciNetCrossRefGoogle Scholar
  17. Semenov, V.S., Korvinski, D.B., Biernat, H.K.: Euler Potentials for the MHD Kamchatnov-Hopf Soliton Solution. Nonlinear Process. Geophys. 9, 347–354 (2002)ADSCrossRefGoogle Scholar
  18. Stern, D.P.: The Motion of Magnetic Field Lines. Space Sci. Rev. 6, 143–173 (1966)ADSCrossRefGoogle Scholar
  19. Webb, G.M.: Multi-Symplectic, Lagrangian, One-Dimensional Gas Dynamics. J. Math. Phys. 56, 053101 (20 pp.) (2015). Also available at
  20. 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). zbMATHGoogle Scholar
  21. 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
  22. 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). ADSCrossRefGoogle Scholar
  23. 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). Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

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

Personalised recommendations