Hamiltonian Approach

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


This chapter describes the Hamiltonian approach to MHD and gas dynamics. In Sect. 8.1 we describe a constrained MHD variational principle by using Lagrange multipliers to enforce the constraints of mass conservation; the entropy advection equation; Faraday’s equation and the so-called Lin constraint describing in part, the vorticity of the flow (i.e. Kelvin’s theorem). This leads to Hamilton’s canonical equations in terms of Clebsch potentials. The Lagrange multipliers define the Clebsch variables, which gives a Clebsch representation for the fluid velocity u (Zakharov and Kuznetsov (1997)). In Sect. 8.2 we transform the canonical, Clebsch variable, Poisson bracket to different non-canonical forms that use Eulerian physical variables (see e.g. Morrison and Greene (1980, 1982), Morrison (1982)) and Holm and Kupershmidt (1983a,b)). The different MHD brackets are described in Sect. 8.3. Section 8.4 verifies the Jacobi identity for the bracket of Morrison and Greene (1982) in which ∇⋅B can be non-zero. We discuss how the Morrison and Greene (1980) bracket, with ∇⋅B = 0, has been placed on a more rigorous footing by the use of the Dirac bracket and projectors by Chandre et al. (2012); Chandre (2013); Chandre et al. (2013) (see also Banerjee and Kumar (2016)). We use the functional multi-vector approach of Olver (1993) to investigate and check the Jacobi identity for: (i) the Morrison and Greene (1982) bracket (ii) the advected A bracket in which A ⋅ dx is Lie dragged with the flow used by (Holm and Kupershmidt (1983a,b)) and (iii) the Morrison and Greene (1980) bracket. The non-canonical Poisson brackets are used to determine the MHD Casimirs in Sect. 8.5 (e.g. Hameiri (2004)). The Casimirs are related to the advected invariants.


  1. Banerjee, R., Kumar, K.: New Approach to Nonrelativistic Ideal Magnetohydrodynamics (2016). arXiv:1601.03944v1 [physics.flu-dyn]Google Scholar
  2. Chandre, C.: Casimir Invariants and the Jacobi Identity in Dirac’s Theory of Constraints of Constrained Hamiltonian Systems. J. Phys. A Math. Theor. 46(37), 375201 (2013)ADSCrossRefzbMATHGoogle Scholar
  3. Chandre, C., Morrison, P.J., Tassi, E.: On the Hamiltonian Formulation of Incompressible Ideal Fluids and Magnetohydrodynamics via Dirac’s Theory of Constraints. Phys. Lett. A 376, 737–743 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. Chandre, C., de Guillebon, L., Back, A., Tassi, E., Morrison, P.J.: On the Use of Projectors for Hamiltonian Systems and Their Relationship with Dirac Brackets. J. Phys. A. Math. Theor. 46(12), 125203 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. Hameiri, E.: The Complete Set of Casimir Constants of the Motion in Magnetohydrodynamics. Phys. Plasmas 11(7), 3423–3431 (2004). ADSMathSciNetCrossRefGoogle Scholar
  6. Holm, D.D., Kupershmidt, B.A.: Poisson Brackets and Clebsch Representations for Magnetohydrodynamics, Multi-Fluid Plasmas and Elasticity. Phys. D 6D, 347–363 (1983a)zbMATHGoogle Scholar
  7. Holm, D.D., Kupershmidt, B.A.: Noncanonical Hamiltonian Formulation of Ideal Magnetohydrodynamics. Physica D 7D, 330–333 (1983b)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler-Lagrange Equations and Semi-products with Application to Continuum Theories. Adv. Math. 137, 1–81 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  10. Marsden, J.E., Ratiu, T., Weinstein, A.: Semidirect Products and Reduction in Mechanics. Trans. Am. Math. Soc. 281(1), 147–177 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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
  12. Morrison, P.J.: Hamiltonian Description of the Ideal Fluid. Rev. Mod. Phys. 70(2), 467–521 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. Morrison, P.J., Greene, J.M.: Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Phys. Rev. Lett. 45, 790–794 (1980)ADSMathSciNetCrossRefGoogle Scholar
  14. Morrison, P.J., Greene, J.M.: Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics (Errata). Phys. Rev. Lett. 48, 569 (1982)ADSCrossRefGoogle Scholar
  15. Olver, P.J.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  16. Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)Google Scholar
  17. Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)ADSGoogle Scholar
  18. Panofsky, W.K.H., Phillips, M.: Classical Electricity and Electromagnetism, sect. 9.4, 2nd edn., p. 164. Wesley, Reading (1964)Google Scholar
  19. Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.: A Solution Adaptive Upwind Scheme for Ideal Magnetohydrodynamics. J. Comput. Phys. 154, 284–309 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. Tur, A.V., Yanovsky, V.V.: Invariants in Dissipationless Hydrodynamic Media. J. Fluid Mech. 248, 67–106 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. Volkov, D.V., Tur, A.V., Janovsky, V.V.: Hidden Supersymmetry of Classical Systems (Hydrodynamics and Conservation Laws). Phys. Lett. A 203, 357–361 (1995)ADSCrossRefGoogle Scholar
  22. 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). ADSGoogle Scholar
  23. Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian Formalism for Nonlinear Waves. Phys. Uspekhi 40(11), 1087–1116 (1997)ADSCrossRefGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

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

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