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A new Hamiltonian formulation for fluids and plasmas. Part 1. The perfect fluid

Published online by Cambridge University Press:  13 March 2009

Jonas Larsson
Jonas Larsson
Affiliation:
Department of Plasma Physics, Umeå University, S-90187 Umeå, Sweden† and Lawrence Berkeley Laboratory, University of California, California 94720, U.S.A.
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Abstract

A new formulation of the Hamiltonian structure underlying the perfect fluid equations is presented. Besides time, a parameter c is also used. Correspondingly, there are two interdependent systems of equations expressing time evolution and e evolution respectively. The accessibility equations define the e dynamics and give the variation in the usual Eulerian fluid variables as determined by the generating functions. The time evolutions of both the Eulerian fluid variables and the generating functions are obtained from an action principle. The consistency of the e and the time dynamics is crucial for this formulation, i.e. the accessibility equations must be propagated in time by the Euler–Lagrange equations. The reason for introducing this new formulation is its power in certain applications where the existing Hamiltonian alternatives seem less convenient to use. In particular, it is a promising tool for Hamiltonian perturbation theory. We consider the small-amplitude expansion, and find, very simply and naturally, the Hermitian structure of the linearized ideal fluid equations as well as coupling coefficients for resonant three-wave interaction exhibiting the Manley–Rowe relations.

Type
Research Article
Information
Journal of Plasma Physics , Volume 55 , Issue 2 , April 1996 , pp. 235 - 259
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Arnold, V. I. 1966 Ann. Inst. Fourier (Grenoble) 16, 319.CrossRefGoogle Scholar
Azizov, T. Ya & Iokhvidov, I. S. 1989 Linear spaces with an Indefinite Metric. Wiley, New York.Google Scholar
Bretherton, F. P. 1970 J. Fluid Mech. 44, 19.CrossRefGoogle Scholar
Brizard, A. 1992 Phys. Lett. 168A, 357.CrossRefGoogle Scholar
Brizard, A., Cook, D. R. & Kaufman, A. N. 1993 Phys. Rev. Lett. 70, 521.CrossRefGoogle Scholar
Bulgakov, A. A., Khankina, S. I. & Yakovenko, V. M. 1971 Soviet Phys. Solid State 12, 1503.Google Scholar
Cendra, H. & Marsden, J. E. 1987 Physica D27, 63.Google Scholar
Dzyaloshinskii, I. E. & Volovick, G. E. 1980 Ann. Phys. (NY) 125, 67.Google Scholar
Eckart, C. 1960 Phys. Fluids. 3, 421.Google Scholar
Elvsén, R. & Larsson, J. 1993 Physica Scripta 47, 571.Google Scholar
Finn, J. M. & Sun, G. 1987 Comments Plasma Phys. Contr. Fusion 11, 7.Google Scholar
Giles, M. J. 1974 Plasma Phys. 16, 99.CrossRefGoogle Scholar
Griffa, A. 1984 Physica A127, 265.CrossRefGoogle Scholar
Herivel, J. W. 1955 Proc. Camb. Phil. Soc. 51, 344.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Phys. Rep. 123, 1.CrossRefGoogle Scholar
Larsson, J. 1982 J. Plasma Phys. 28, 215.Google Scholar
Larsson, J. 1991 Phys. Rev. Lett. 66, 1466.CrossRefGoogle Scholar
Larsson, J. 1992 J. Plasma Phys. 48, 13.Google Scholar
Larsson, J. 1993 J. Plasma Phys. 49, 255.CrossRefGoogle Scholar
Larsson, J. 1996 a J. Plasma Phys. 55, 261.CrossRefGoogle Scholar
Larsson, J. 1996b J. Plasma Phys. 55, 279.CrossRefGoogle Scholar
Larsson, J. & Stenflo, L. 1973 Beitr. Plasmaphys. 14, 7.Google Scholar
Lin, C. C. 1963 Hydrodynamics of helium II. Proceedinys XXI International School of Physics, pp. 93–146. Academic Press, New York.Google Scholar
Lindoren, T. 1982 Physica Scripta 25, 568.Google Scholar
Lindgren, T., Larsson, J. & Stenflo, L. 1981 J. Plasma Phys. 26, 407.CrossRefGoogle Scholar
Louisell, W. H. 1960 Coupled Mode and Parametric Electronics. Wiley, New York.Google Scholar
Manley, J. M. & Rowe, H. E. 1956 Proc. IRE 44, 904.Google Scholar
Marsden, J. F. & Ratiu, T. S. 1994 Anlntroduction. to Mechanics and Symmetry. Springer- Verlag, New York.Google Scholar
Marsden, J. E. & Scheurle, J. 1993 The reduced Euler–Lagrange equations. Fields Inst. Commun. (A.MS) 1, 139.Google Scholar
Marsden, J. E., Ratiu, T. S. & Weinstein, A. 1984 Trans. Am. Math. Soc. 291, 147.CrossRefGoogle Scholar
Mobbs, S. D. 1982 Proc. R. Soc. Lond. A381, 457.Google Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (ed. Tabor, M. & Treve, Y. M.), p. 13. AlP Conf. Proc., Vol. 88, AlP, New York.Google Scholar
Morrison, P. J. & Greene, J. M. 1980 Phys. Rev. Lett. 45, 790.CrossRefGoogle Scholar
Morrison, P. J. & Greene, J. M. 1982 Phys. Rev. Lett. 48, 569.Google Scholar
Morrison, P. J. & Pforsch, D. 1990 Phys. Fluids B2, 1105.CrossRefGoogle Scholar
Morrison, P. J. & Pflrsch, D. 1992 Phys. Fluids B4, 3038.Google Scholar
Newcomb, W. A. 1962 Fusion Suppl. Part 2, 451.Google Scholar
Phillips, G. M. 1974 Nonlinear Waves (ed. Leibovich, S. & Seebass, A. R.). Cornell University Press, Ithaca.Google Scholar
Salmon, R. 1983 J. Fluid Mech. 132, 431.Google Scholar
Salmon, R. 1985 J. Fluid Mech. 153, 461.Google Scholar
Salmon, R. 1988 a J. Fluid Mech. 196, 345.CrossRefGoogle Scholar
Salmon, R. 1988b Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20, 225.Google Scholar
Seliger, R. L. & Whitham, G. B. 1968 Proc. R. Soc. Lond. A305, 1.Google Scholar
Sfrrin, J. 1959 Mathematical principles of classical fluid mechanics. Hand buch der Physik VIII -I, pp. 125–263. Springer-Verlag, Berlin.Google Scholar
Simmons, W. F. 1969 Proc. R. Soc. Lond. A309, 551.Google Scholar
Sjülund, A. & Stenflo, L. 1967 Physica 35, 499.Google Scholar
Stenflo, L. 1994 Physica Scripta T50, 15.CrossRefGoogle Scholar
Van, Saarloos W. 1981 Physica A108, 557.Google Scholar
Weiland, J. & Wilhelmsson, H. 1977 Coherent Non-Linear Interaction of Waves in Piasmas. Pergamon Press, Oxford.Google Scholar
Yeh, K. C. & Liu, C. H. 1981 J. Geophys. Res. 86, 9722.Google Scholar
Yeh, K. C. & Liu, C. H., 1985 Radio Sci. 20, 1279.CrossRefGoogle Scholar