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Multi-Symplectic Clebsch Approach

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

Abstract

Multi-symplectic formulations of Hamiltonian systems with two or more independent variables x α have been developed as a useful extension of Hamiltonian systems with one evolution variable t. This development has connections with dual variational formulations of traveling wave problems (e.g. Bridges (1992)), and is useful in numerical schemes for multisymplectic systems. Bridges and co-workers used the multi-symplectic approach to study linear and nonlinear wave propagation, generalizations of wave action, wave modulation theory, and wave stability problems (Bridges (1997a,b)). Reich (2000), and Bridges (2006) develop difference schemes. Multi-symplectic Hamiltonian systems have been studied by Marsden and Shkoller (1999) and Bridges et al. (2005). Bridges et al. (2010) shows the connection between multi-symplectic systems and the variational bi-complex.

References

  1. Boillat, G.: Simple Waves in N-Dimensional Propagation. J. Math. Phys. 11, 1482–1483 (1970)ADSCrossRefzbMATHGoogle Scholar
  2. Bridges, T.J.: Spatial Hamiltonian Structure, Energy Flux and the Water Wave Problem. Proc. R. Soc. Lond. A 439, 297–315 (1992)CrossRefzbMATHGoogle Scholar
  3. Bridges, T.J.: Multi-Symplectic Structures and Wave Propagation. Math. Proc. Camb. Philos. Soc. 121, 147–190 (1997a)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bridges, T.J.: A Geometric Formulation of the Conservation of Wave Action and Its Implications for Signature and Classification of Instabilities. Proc. R. Soc. A 453, 1365–1395 (1997b)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. Bridges, T.J., Reich, S.: Numerical Methods for Hamiltonian PDEs. J. Phys. A Math. Gen. 39(22), 5287–5320 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. Bridges, T.J., Hydon, P.E., Reich, S.: Vorticity and Symplecticity in Lagrangian Fluid Dynamics. J. Phys. A Math. Gen. 38, 1403–1418 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. Bridges, T.J., Hydon, P.E., Lawson, J.K.: Multi-Symplectic Structures and the Variational Bi-Complex. Math. Proc. Camb. Philos. Soc. 148, 159–178 (2010)CrossRefzbMATHGoogle Scholar
  8. Brio, M., Zakharian, A.R., Webb, G.M.: Numerical Time-Dependent Partial Differential Equations for Scientists and Engineers. In: Chui, C.K. (ed.) Mathematics in Science and Engineering 123, 1st edn. Elsevier, Burlington (2010)Google Scholar
  9. Carenina, J.F., Crampin, M., Ibort, L.A.: On the Multisymplectic Formalism for First Order Field Theories. In: Differential Geometry and Its Applications 1, pp. 345–374. North Holland, Amsterdam (1991)Google Scholar
  10. Cotter, C.J., Holm, D.D., Hydon, P.E.: Multi-Symplectic Formulation of Fluid Dynamics Using the Inverse Map. Proc. R. Soc. Lond. 463, 2617–2687 (2007)CrossRefzbMATHGoogle Scholar
  11. Forger, M., Paufler, C., Römer, H.: A General Construction of Poisson Brackets on Exact Multisymplectic Manifolds. Rep. Math. Phys. 51, 187–195 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Gotay, M.J.: A Multisymplectic Framework for Classical Field Theory and the Calculus of Variations II: Space+Time Decomposition. Differ. Geom. Appl. 1, 375–390 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gotay, M.J., Isenberg, J., Marsden, J.E., Montgomery, R. (with Sniatycki, J., Yasskin, P.B., collaborators): Momentum Maps and Classical Fields, Part I: Covariant Field Theory (2004a). arXiv:physics/9801019v2[math-ph]Google Scholar
  14. Gotay, M.J., Isenberg, J., Marsden, J.E. (with Montgomery, R., Sniatycki, J., Yasskin, P.B. collaborators): Momentum Maps and Classical Fields, Part II: Canonical Analysis of Field Theories (2004b). arXiv:math-ph/0411032v1Google Scholar
  15. Grundland, A.M., Picard, P.: On Conditionally Invariant Solutions of Magnetohydrodynamic Equations Multiple Waves. J. Nonlinear Math. Phys. 11(1), 47–74 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. Harrison, B.K., Estabrook, F.B.: Geometric Approach to Invariance Groups and Solution of Partial Differential Systems. J. Math. Phys. 12, 653–666 (1971)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. Hydon, P.E.: Multi-Symplectic Conservation Laws for Differential and Differential-Difference Equations. Proc. R. Soc. A 461, 1627–1637 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. Kanatchikov, I.V.: On the Canonical Structure of the de-Donder-Weyl Covariant Hamiltonian Formulation of Field Theory I. Graded Poisson Brackets and Equations of Motion (1993). Preprint, arXiv:hep-th/9312162v1Google Scholar
  19. Kanatchikov, I.V.: On Field Theoretic Generalizations of a Poisson Algebra. Rep. Math. Phys. 40, 225–234 (1997). hep-th/9710067Google Scholar
  20. Kanatchikov, I.V.: Canonical Structure of Classical Field Theory in the Polymomentum Phase Space. Rep. Math. Phys. 41, 49–90 (1998). hep-th/9709229Google Scholar
  21. Marsden, J.E., Shkoller, S.: Multi-Symplectic Geometry, Covariant Hamiltonians and Water Waves. Math. Proc. Camb. Philos. Soc. 125, 553–575 (1999)CrossRefzbMATHGoogle Scholar
  22. Marsden, J.E., Pekarsky, S., Shkoller, S., West, M.: Variational Methods, Multisymplectic Geometry, and Continuum Mechanics. J. Geom. Phys. 38, 253–284 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. Reich, S.: Multi-Symplectic Runge-Kutta Collocation Methods for Hamiltonian Wave Equations. J. Comput. Phys. 157, 473 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. Webb, G.M.: Multi-Symplectic, Lagrangian, One-Dimensional Gas Dynamics. J. Math. Phys. 56, 053101 (20 pp.) (2015). Also available at http://arxiv.org/abs/1408.4028v4
  25. Webb, G.M., Anco, S.C.: Vorticity and Symplecticity in Multi-Symplectic, Lagrangian Gas Dynamics. J. Phys. A Math. Theor. 49, 075501 (44 pp.) (2016). https://doi.org/10.1008/1751-8113/49/075501
  26. Webb, G.M., Ratkiewicz, R., Brio, M., Zank, G.P.: Multi-Dimensional MHD Simple Waves. In: Winterhalter, D., Gosling, J.T., Habbal, S.R., Kurth, W.S., Neugebauer, M. (eds.) Proceedings of the 8th International Solar Wind Conference: Solar Wind Eight. AIP Conference Proceedings 382, pp. 335–338. AIP, New York (1996)Google Scholar
  27. Webb, G.M., Hu, Q., Dasgupta, B., Roberts, D.A., Zank, G.P.: Alfven Simple Waves: Euler Potentials and Magnetic Helicity. Astrophys. J. 725, 2128–2151 (2010b). https://doi.org/10.1088/0004-637X/725/2/2128 ADSCrossRefGoogle Scholar
  28. Webb, G.M., Zank, G.P., Burrows, R.H., Ratkiewicz, R.E.: Simple Alfven Waves. J. Plasma Phys. 77(Part 1), 51–93 (2011). https://doi.org/10.101/S00233377809990596
  29. Webb, G.M., Hu, Q., Dasgupta, B., Zank, G.P.: Double Alfvén Waves. J. Plasma Phys. 78(Part 1), 71–85 (2012a). https://doi.org/10.1017/S0022377811000420
  30. Webb, G.M., Dasgupta, B., McKenzie, J.F., Hu, Q., Zank, G.P.: Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics I: Lie Dragging Approach. J. Phys. A. Math. Theor. 47, 095501 (33 pp.) (2014a). https://doi.org/10.1088/1751-8113/49/9/095501. Preprint available at http://arxiv.org/abs/1307.1105
  31. Webb, G.M., Dasgupta, B., McKenzie, J.F., Hu, Q., Zank, G.P.: Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether’s Theorems and Casimirs. J. Phys. A. Math. Theor. 47, 095502 (31 pp.) (2014b). https://doi.org/10.1088/1751-8113/47/9/095502. Preprint available at http://arxiv.org/abs/1307.1038
  32. Webb, G.M., McKenzie, J.F., Zank, G.P.: Multi-Symplectic Magnetohydrodynamics. J. Plasma Phys. 80(Part 5), 707–743 (2014c). https://doi.org/10.1017/S0022377814000257. Also at http://arxiv.org/abs/1312.4890v4
  33. Webb, G.M., Burrows, R.H., Ao, X., Zank, G.P.: Ion Acoustic Travelling Waves. J. Plasma Phys. 80(Part 2), 147–171 (2014d). https://doi.org/10.1017/S0022377813001013. Preprint at http://arxiv.org/abs/1312.6406
  34. Webb, G.M., McKenzie, J.F., Zank, G.P.: Multi-Symplectic Magnetohydrodynamics: II, Addendum and Erratum. J. Plasma Phys. 81, 905810610 (15 pp.) (2015). https://doi.org/10.1017/S0022377815001415
  35. Yoshida, Z.: Clebsch Parameterization: Basic Properties and Remarks on Its Applications. J. Math. Phys. 50, 113101 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian Formalism for Nonlinear Waves. Phys. Uspekhi 40(11), 1087–1116 (1997)ADSCrossRefGoogle 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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