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Euler-Poincaré Equation Approach

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

Abstract

Poincaré (1901) wrote down the Euler equations for a rigid body on so(3) in a matrix commutator form (see also Holm (2008b), Volume 2, p. 46). Arnold (1966) showed that the equations for ideal, incompressible fluid dynamics could be derived from a variational principle in which the Lagrangian consists of the fluid kinetic energy, subject to an infinite Lie group (pseudo-Lie group) constraint, associated with the Lagrangian map (the constraint is that the Lagrangian map x = φ(x0, t) for fixed t, is a differentiable (smooth) and measure preserving diffeomorphism). The group G, is known as Sdiff(R3). The variational formulation showed that when the Lagrangian l is a metric on the tangent space TG, the resultant variational equations (the Euler-Poincaré equations) are geodesic spray equations for geodesic motion on the group G with respect to the metric l. For the case of rigid body dynamics the group involved is the semi-direct product Lie group SE(3) = SO(3)ⓈR3. Euler-Poincaré variational principles have been developed by a number of authors (e.g. Marsden et al. (1984), Holm and Kupershmidt (1983a,b), Holm et al. (1998),Cendra et al. (2003), Arnold and Khesin (1998)). The geodesic spray equations for MHD were obtained by Ono (1995a,b). These equations are sometimes referred to as the Euler-Arnold equations. Araki (2015, 2017) determine the geodesic spray equations for incompressible Hall plasmas, known as XMHD (i.e. extended MHD). The curvature associated with the geodesic metric is negative for unstable flows. Holm et al. (1985) describes the use of Casimirs in stability analyses.

References

  1. Araki, K.: Differential-Geometrical Approach to the Dynamics of Dissipationless, Incompressible Hall MHD: I. Lagrangian Mechanics on Semi-direct Products of Two Volume Preserving Diffeomorphisms and Conservation Laws. J. Phys. A: Math. Theor. 48, 175501 (2015)Google Scholar
  2. Araki, K.: Differential-Geometrical Approach to the Dynamics of Dissipationless Incompressible Hall Magnetohydrodynamics: II, Geodesic Formulation and Riemannian Curvature Analysis of Hydrodynamic and Magnetohydrodynamic Stabilities. J. Phys. A 50, 235501 (32 pp.) (2017)Google Scholar
  3. Arnold, V.I.: Sur la geometrie differentielle des groups de Lie de dimension infinie et ses applications á l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier Grenoble 16, 319–361 (1966)Google Scholar
  4. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, New York (1998)Google Scholar
  5. Cendra, H., Marsden, J.E., Pekarsky, S., Ratiu, T.S.: Variational Principles for Li-Poisson and Hamilton-Poincaré Equations. Mosc. Math. J. 3(3), 837–867 (2003)Google Scholar
  6. Cotter, C.J., Holm, D.D.: On Noether’s Theorem for Euler Poincaré Equation on the Diffeomorphism Group with Advected Quantities. Found. Comput. Math. (2012). https://doi.org/10.1007/S10208-012-9126-8
  7. Holm, D.D.: Geometric Mechanics, Part II: Rotating, Translating and Rolling. Imperial College Press, London (2008b). Distributed by World ScientificGoogle Scholar
  8. Holm, D.D., Kupershmidt, B.A.: Poisson Brackets and Clebsch Representations for Magnetohydrodynamics, Multi-Fluid Plasmas and Elasticity. Phys. D 6D, 347–363 (1983a)Google Scholar
  9. Holm, D.D., Kupershmidt, B.A.: Noncanonical Hamiltonian Formulation of Ideal Magnetohydrodynamics. Physica D 7D, 330–333 (1983b)Google Scholar
  10. 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). https://doi.org/0370-1573/85
  11. 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)Google Scholar
  12. Hydon, P.E., Mansfield, E.L.: Extensions of Noether’s Second Theorem: From Continuous to Discrete Systems. Proc. R. Soc. A 467, 3206–3221 (2011). https://doi.org/https://doi.org/10.1098/rspa.2011.0158Google Scholar
  13. Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Dordrecht (1985)Google Scholar
  14. Kruskal, M.D., Kulsrud, R.M.: Equilibrium of a Magnetically Confined Plasma in a Toroid. Phys. Fluids 1, 265 (1958)Google Scholar
  15. Marsden, J.E., Ratiu, T., Weinstein, A.: Semidirect Products and Reduction in Mechanics. Trans. Am. Math. Soc. 281(1), 147–177 (1984)Google Scholar
  16. Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W.H. Freeman, San Francisco (1973)Google Scholar
  17. Moffatt, H.K.: The Degree of Knottedness of Tangled Vortex Lines. J. Fluid. Mech. 35, 117 (1969)Google Scholar
  18. Moffatt, H.K.: Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, Cambridge (1978)Google Scholar
  19. Moffatt, H.K., Ricca, R.L.: Helicity and the Calugareanu Invariant. Proc. R. Soc. Lond. Ser. A 439, 411 (1992)Google Scholar
  20. Newcomb, W.A.: Lagrangian and Hamiltonian Methods in Magnetohydrodynamics. Nucl. Fusion Suppl. (Part 2), 451–463 (1962)Google Scholar
  21. Ono, T.: Riemannian Geometry of an Ideal Incompressible Magnetohydrodynamical Fluid. Physica D 81, 207–220 (1995a)Google Scholar
  22. Ono, T.: A Riemannian Geometrical Description for Lie-Poisson Systems and Its Application to Idealized Magnetohydrodynamics. J. Phys. A 28, 1737–1651 (1995b)Google Scholar
  23. 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
  24. Padhye, N.S., Morrison, P.J.: Fluid Relabeling Symmetry. Phys. Lett. A 219, 287–292 (1996a)Google Scholar
  25. Padhye, N.S., Morrison, P.J.: Relabeling Symmetries in Hydrodynamics and Magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877 (1996b)ADSGoogle Scholar
  26. Poincaré, H.: Sue une forme nouvelle des equations de la mechanique. C.R. Acad. Sci. 132, 369–371 (1901)Google Scholar
  27. Squire, J., Qin, H., Tang, W.M., Chandre, C.: The Hamiltonian Structure and Euler-Poincaré Formulation of the Vlasov-Maxwell and Gyrokinetic Systems. Phys. Plasmas 20, 122501 (14 pp.) (2013)Google Scholar
  28. Tur, A.V., Yanovsky, V.V.: Invariants in Dissipationless Hydrodynamic Media. J. Fluid Mech. 248, 67–106 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. Webb, G.M., Mace, R.L.: Potential Vorticity in Magnetohydrodynamics. J. Plasma Phys. 81, p. 18, 905810115 (2015). https://doi.org/10.1017/S0022377814000658. Preprint: http://arxiv/org/abs/1403.3133
  30. 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
  31. 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
  32. 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
  33. Woltjer, L.: A Theorem on Force-Free Magnetic Fields. Proc. Natl. Acad. Sci. 44, 489 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

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

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