Abstract
A Hamiltonian formulation of hydrodynamics is well known for the case of purely irrotational flows and it now exists in a more general case as well. The minimal extension of the action principle is obtained from two axioms: 1. That the number of independent degrees of freedom be 4, as in standard hydrodynamics. 2. That the equation of continuity must be one of the Euler-Lagrange equations, and that it allows for vorticity. Applications include: 1. Couette flow with a new criterion for the breakdown of laminar motion. 2. A rotating source for Einstein’s equation that respects the Bianchi identity. 3. A new approach to the electromagnetism of fluids. 4. A rigorous virial theorem for fluids. 5. A critique of the current state of the theory of atmospheres.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. D. Andereck, S. S. Liu, and H. L. Swinney, “Flow regimes in a circular Couette system with independently rotating cylinders”, J. Fluid Mech. 164, 155–183 (1986).
A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua (New York: MacGraw-Hill, 1980).
G. K. Batchelor, “On the spontaneous magnetic field in a conducting liquid in turbulent motion”, Proc. R. Soc. A, 405 (1949).
A. Einstein, “On the electrodynamics of moving bodies”, Ann. Phys. (1905).
A. Einstein and J. Laub, “Uber die electromagnetischen Grundgleichungen für bewegte Körper”, Ann. Phys. 26, 532–540 (1908a).
A. Einstein and J. Laub, “Uber die im electromagnetischen Felde af ruhende Körper ausgebten ponderomotorischen Kräfte”, Ann. Phys. 14, 327–336 (1908b).
A. L. Fetter, “Rotating trapped Bose-Einstein condensates”, Rev. Mod. Phys. 81, 647–691 (2009).
R. P. Feynman, “Atomic theory of the two-fluid model of liquid helium”, Phys. Rev. 94, 262–277 (1954).
C. Fronsdal, “Ideal stars and general relativity”, Gen. Rel. Grav. 39, 1971–2000 (2007a).
C. Fronsdal, Action Principle for Hydrodynamics and Thermodynamics including general, rotational flows, arxiv1405.7138v3[physics.gen-ph](v3 19 June 2015).
J. W. Gibbs, “On the equilibrium of heterogeneous substances”, Trans. Conn. Acad. 3, 108–524 (1878).
H. E. Hall and W. F. Vinen, “The Rotation of Liquid Helium II. The Theory of Mutual Friction in Uniformly Rotating Helium II”, Proc. R. Soc. Lond. A, 238 (1956). doi 10.1098/rspa.1956.0215
M. Kalb and P. Ramond, Phys. Rev. D 9, 2273 (1974).
H. A. Lorentz, “La Théorie” electromagnétique de Maxwell et son application aux corps mouvants”, Arch. Néerlandaises des Sci. Exactes et Naturelles 25, 363–552 (1892).
H. Lorentz, Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern (E.J. Brill, Leiden, 1895).
H. A. Lorentz, Lectures on Theoretical Physics (Macmillan, London, 1927–1931), Vol. 3, p. 303.
I. M. Khalatnikov, An Introduction to the Theory of Superfluidity (Westview Press, 2000).
L. D. Landau and E. M. Lifschitz, Dokl. Akad. Nauk 100, 669 (1955).
L. D. Landau and E. M. Lifschitz, Statistical Physics (Pergamon Press, 1958).
H. Martin. doi1115/IHTC 14-22023
J. C. Maxwell, M. Rukeyser, Willard Gibbs (OX BOW PReSS, Woodbridge, Conn., 1942), p. 199.
H. Minkowski, Gött. Nachr., 45 (1908).
V. I. Ogievetskij and I. V. Polubarinov, “Minimal interactions between spin 0 and spin 1 fields”, J. Exptl. Theor. Phys. (USSR) 46, 1048–1055 (1964).
G. N. Pellegrini and A. R. Swift, “Maxwell’s equations in a rotating medium: Is there a problem?”, Am. J. Phys. 63, 694–705 (1995).
H. Poincaré, Thermodynamique (Gauthier-Villars, Paris, 1908).
R. L. Scott and P. H. van Konynenburg, Roy. Soc. Lond. A 298, 495–540 (1980).
S. Putterman, Superfluid Hydrodynamics (North-holland, 1974).
R. C. Tolman, “The electromotive force produced in solutions by centrifugal action”, M.I.T. Res. Lab. Contributions, No. 59 (1910).
R. C. Tolman, Proc. Am. Acad. 46, 284 (1910).
R. C. Tolman and T. D. Stewart, “The electromotive force produced by acceleration of metals”, Phys. Rev. 8, 97–116 (1916).
R. C. Tolman, “Osgerby and Stewart”, J. Am. Chem. Soc. 36, 466 (1914).
M. A. Virasoro, “Variational principle for two-dimensional incompressible hydro-dynamics and quasiostrophic flows”, Phys. Rev. Lett. 47, 1181–1183 (1981).
H. A. Wilson, “On the electric effect of rotating a dielectric in a magnetic field”, Phil. Trans. A 204 (1904).
M. Wilson and H. A. Wilson, “On the electric effect of rotating a magnetic insulator in a magnetic field”, Proc. R. Soc. Lond., Ser. A 89, 99–106 (1913).
A. Zheltukhin, On Brane Symmetries, arXiv.1409.6655.
Author information
Authors and Affiliations
Corresponding author
Additional information
For Special Issue Dedicated to the Memory of V.G. Kadyshevsky
The article is published in the original.
Rights and permissions
About this article
Cite this article
Fronsdal, C. Action principles for hydro- and thermo-dynamics. Phys. Part. Nuclei 48, 211–226 (2017). https://doi.org/10.1134/S1063779617020046
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063779617020046