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Euler-Poincaré Dynamics of Perfect Complex Fluids

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Geometry, Mechanics, and Dynamics

Abstract

Lagrangian reduction by stages is used to derive the Euler-Poincaré equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geometrically either as objects in a vector space, or as coset spaces of Lie symmetry groups with respect to subgroups that leave these objects invariant. Examples include liquid crystals, superfluids, Yang-Mills magnetofluids and spin-glasses. A Lie-Poisson Hamiltonian formulation of the dynamics for perfect complex fluids is obtained by Legendre transforming the Euler-Poincaré formulation. These dynamics are also derived by using the Clebsch approach. In the Hamiltonian and Lagrangian formulations of perfect complex fluid dynamics Lie algebras containing two-cocycles arise as a characteristic feature.

After discussing these geometrical formulations of the dynamics of perfect complex fluids, we give an example of how to introduce defects into the order parameter as imperfections (e.g., vortices) that carry their own momentum. The defects may move relative to the Lagrangian fluid material and thereby produce additional reactive forces and stresses.

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References

  • Balatskii, A. [1990], Hydrodynamics of an antiferromagnet with fermions, Phys. Rev. B 42, 8103–8109.

    Article  Google Scholar 

  • Beris, A. N. and B. J. Edwards [1994], Thermodynamics of Flowing Systems with internal microstructure, Oxford University Press.

    Google Scholar 

  • Cendra, H., D. D. Holm, J. E. Marsden and T. S. Ratiu [1999], Lagrangian Reduction, the Euler-Poincaré Equations, and Semidirect Products. Arnol’d Festschrift Volume II, 186, Amer. Math. Soc. Transl. Ser. 2, pp. 1–25.

    MathSciNet  Google Scholar 

  • Cendra, H., J. E. Marsden, and T. Ratiu [2001], Lagrangian Reduction by Stages. Mem. Amer. Math. Soc. 152, no. 722, viii+108 pp.

    Google Scholar 

  • Chandrasekhar, S. [1992], Liquid Crystals, Second Edition. Cambridge University Press, Cambridge.

    Google Scholar 

  • Coquereaux, R. and A. Jadcyk [1994], Riemann Geometry Fiber Bundles Kaluza-Klein Theories and all that..., World Scientific, Lecture Notes in Physics, vol. 16.

    Google Scholar 

  • Cosserat, E. and F. Cosserat [1909], Théorie des corps deformable. Hermann, Paris.

    Google Scholar 

  • de Gennes, P. G. and J. Prost [1993], The Physics of Liquid Crystals, Second Edition. Oxford University Press, Oxford.

    Google Scholar 

  • Dunn, J. E. and J. Serrin [1985], On the thermodynamics of interstitial working, Arch. Rat. Mech. Anal. 88, 95–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Dzyaloshinskii, I. E. [1977], Magnetic structure of UO2, Commun. on Phys. 2, 69–71.

    Google Scholar 

  • Dzyaloshinskii, I. E. and G. E. Volovick [1980], Poisson brackets in condensed matter physics, Ann. of Phys. 125, 67–97.

    Article  MathSciNet  Google Scholar 

  • Ericksen, J. L. [1960], Anisotropic fluids, Arch. Rational Mech. Anal. 4, 231–237.

    Article  MATH  MathSciNet  Google Scholar 

  • Ericksen, J. L. [1961], Conservation laws for liquid crystals, Trans. Soc. Rheol. 5, 23–34.

    Article  MathSciNet  Google Scholar 

  • Eringen, A. C. [1997], A unified continuum theory of electrodynamics of liquid crystals, Internat. J. Engrg. Sci. 35, 1137–1157.

    Article  MATH  MathSciNet  Google Scholar 

  • Flanders, H. [1989], Differential Forms with Applications to the Physical Sciences, Dover Publications: New York.

    MATH  Google Scholar 

  • Fuller, F. B. [1978], Decomposition of linking number of a closed ribbon: problem from molecular-biology, Proc. Nat. Acad. Sci. USA 75, 3557–3561.

    MATH  MathSciNet  Google Scholar 

  • Gibbons, J., D. D. Holm and B. Kupershmidt [1982], Gauge-invariant Poisson brackets for chromohydrodynamics, Phys. Lett. A 90, 281–283.

    Article  MathSciNet  Google Scholar 

  • Gibbons, J., D. D. Holm and B. Kupershmidt [1983], The Hamiltonian structure of classical chromohydrodynamics, Physica D 6, 179–194.

    Article  MathSciNet  Google Scholar 

  • Goldstein, R. E., T. R. Powers and C. H. Wiggins [1998], Viscous nonlinear dynamics of twist and writhe, Phys. Rev. Lett. 80, 5232–5235.

    Article  Google Scholar 

  • Golo, V. L. and M. I. Monastyrskii [1977], Topology of gauge fields with several vacuums, JETP Lett. 25, 251–254. [Pis’ma Zh. Eksp. Teor. Fiz. 25, 272–276.]

    Google Scholar 

  • Golo, V. L. and M. I. Monastyrskii [1978], Currents in superfluid 3He, Lett. Math. Phys. 2, 379–383.

    Article  MathSciNet  Google Scholar 

  • Golo, V. L., M. I. Monastyrskii and S. P. Novikov [1979], Solutions of the Ginzburg-Landau equations for planar textures in superfluid 3He, Comm. Math. Phys. 69, 237–246.

    Article  MathSciNet  Google Scholar 

  • Goriely, A. and M. Tabor [1997], Nonlinear dynamics of filaments. 1. Dynamical instabilities, Phys. D 105, 20–44.

    Article  MathSciNet  MATH  Google Scholar 

  • Hall, H. E. [1985], Evidence for intrinsic angular momentum in superfluid 3He-A, Phys. Rev. Lett. 54, 205–208.

    Article  Google Scholar 

  • Hohenberg, P.C. and B. I. Halperin [1977], Theory of dynamical critical phenomena, Rev. Mod. Phys. 49, 435–479.

    Article  Google Scholar 

  • Holm, D. D. [1987], Hall magnetohydrodynamics: conservation laws and Lyapunov stability, Phys. Fluids 30, 1310–1322.

    Article  MATH  MathSciNet  Google Scholar 

  • Holm, D. D. [2001], Introduction to HVBK dynamics, in Quantized Vortex Dynamics and Superfluid Turbulence. (C. F. Barenghi, R. J. Donnelly and W. F. Vinen, eds.) Lecture Notes in Physics, volume 571, Springer-Verlag, pp. 114–130.

    Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1982], Poisson structures of superfluids, Phys. Lett. A 91, 425–430.

    Article  MathSciNet  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1983a], Poisson structures of superconductors, Phys. Lett. A 93, 177–181.

    Article  MathSciNet  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1983b], Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Physica D 6, 347–363.

    Article  MathSciNet  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1984], Yang-Mills magnetohydrodynamics: nonrelativistic theory, Phys. Rev. D 30, 2557–2560.

    Article  MathSciNet  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1986], Hamiltonian structure and Lyapunov stability of a hyperbolic system of two-phase flow equations including surface tension, Phys. Fluids 29, 986–991.

    Article  MathSciNet  MATH  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1987], Superfluid plasmas: Multivelocity nonlinear hydrodynamics of superfluid solutions with charged condensates coupled electromagnetically, Phys. Rev. A 36, 3947–3956.

    Article  Google Scholar 

  • Holm, D. D. and B. A. Kupershmidt [1988], The analogy between spin glasses and Yang-Mills fluids, J. Math. Phys. 29, 21–30.

    Article  MathSciNet  MATH  Google Scholar 

  • Holm, D. D., J. E. Marsden, and T. S. Ratiu [1998], The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math. 137, 1–81.

    Article  MathSciNet  MATH  Google Scholar 

  • Isaev, A. A., M. Yu. Kovalevskii, and S. V. Peletminskii [1995], Hamiltonian appraoch to continuum dynamic, Theoret. and Math. Phys. 102, 208–218. [Teoret. Math. Fiz. 102, 283–296.]

    Article  MathSciNet  MATH  Google Scholar 

  • Isayev, A. A., M. Yu. Kovalevsky and S. V. Peletminsky [1997], Hydrodynamic theory of magnets with strong exchange interaction, Low Temp. Phys. 23, 522–533.

    Article  Google Scholar 

  • Isayev, A. A. and S. V. Peletminsky [1997]. On Hamiltonian formulation of hydrodynamic equations for superfluid 3He-3, Low Temp. Phys. 23, 955–961.

    Article  Google Scholar 

  • Jackiw, R. and N. S. Manton [1980], Symmetries and conservation laws in gauge theories, Ann. Phys. 127, 257–273.

    Article  MathSciNet  Google Scholar 

  • Kamien, R. D. [1998], Local writhing dynamics, Eur. Phys. J. B 1, 1–4.

    Article  Google Scholar 

  • Kats, E. I. and V. V. Lebedev [1994], Fluctuational E ects in the Dynamics of Liquid Crystals, Springer: New York.

    Google Scholar 

  • Khalatnikov, I. M. and V. V. Lebedev [1978], Canonical equations of hydrody-namics of quantum liquids, J. Low Temp. Phys. 32, 789–801

    Article  Google Scholar 

  • Khalatnikov, I. M. and V. V. Lebedev [1980], Equation of hydrodynamics of quantum liquid in the presence of continuously distributed singular solitons, Prog. Theo. Phys. Suppl. 69, 269–280.

    Google Scholar 

  • Klapper, I. [1996], Biological applications of the dynamics of twisted elastic rods, J. Comp. Phys. 125, 325–337.

    Article  MATH  MathSciNet  Google Scholar 

  • Kleinert, H. [1989], Gauge Fields in Condensed Matter, Vols. I, II, World Scientific.

    Google Scholar 

  • Kléeman, M. [1983], Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media, John Wiley and Sons.

    Google Scholar 

  • Kléeman, M. [1989], Defects in liquid crystals, Rep. on Prog. in Phys. 52, 555–654.

    Article  MathSciNet  Google Scholar 

  • Kuratsuji, H. and H. Yabu [1998], Force on a vortex in ferromagnet model and the properties of vortex configurations, J. Phys. A 31, L61–L65.

    Article  MathSciNet  MATH  Google Scholar 

  • Lammert, P. E., E. S. Rokhsar and J. Toner [1995], Topological and nematic ordering. I. A gauge theory, Phys. Rev. E 52, 1778–1800.

    Article  MathSciNet  Google Scholar 

  • Leggett, A. J. [1975], A theoretical description of the new phases of 3He, Rev. Mod. Phys. 47, 331–414.

    Article  Google Scholar 

  • Leslie, F. M. [1966], Some constitutive equations for anisotropic fluids, Quart. J. Mech. Appl. Math. 19, 357–370.

    MATH  MathSciNet  Google Scholar 

  • Leslie, F. M. [1968], Some constitutive equations for liquid crystals, Arch. Rational Mech. Anal. 28, 265–283.

    Article  MATH  MathSciNet  Google Scholar 

  • Leslie, F. M. [1979], Theory of flow phenomena in liquid crystals, in Advances in Liquid Crystals, vol. 4, (G. H. Brown, ed.) Academic, New York pp. 1–81.

    Google Scholar 

  • Marsden, J. E. and T. S. Ratiu [1999], Introduction to Mechanics and Symmetry, Second Edition, Springer-Verlag, Texts in Applied Mathematics 17.

    Google Scholar 

  • Marsden, J. E., T. S. Ratiu and J. Scheurle [2000], Reduction theory and the Lagrange-Routh equations, J. Math. Phys. 41, 3379–3429.

    Article  MathSciNet  MATH  Google Scholar 

  • Marsden, J. E. and J. Scheurle [1995], The Lagrange-Poincarée equations, Fields Institute Commun. 1, 139–164.

    MathSciNet  Google Scholar 

  • Marsden, J. E. and A. Weinstein [1974], Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121–130.

    Article  MathSciNet  MATH  Google Scholar 

  • Marsden, J. E. and A. Weinstein [1983], Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D 7, 305–323.

    Article  MathSciNet  Google Scholar 

  • Mermin, N. D. [1979], The topological theory of defects in ordered media, Rev. Mod. Phys. 51, 591–648.

    Article  MathSciNet  Google Scholar 

  • Mermin, N. D. and T.-L. Ho [1976], Circulation and angular momentum in the A phase of superfluid Helium-3, Phys. Rev. Lett. 36, 594–597.

    Article  Google Scholar 

  • Mineev, V. P. [1980], Topologically stable defects and solitons in ordered media, Soviet Science Reviews, Section A: Physics Reviews, vol. 2, (I. M. Khalatnikov, ed.) (Chur, London, New York: Harwood Academic Publishers) pp. 173–246.

    Google Scholar 

  • Olver, P. J. [1993], Applications of Lie groups to di erential equations, Second Edition, Springer-Verlag, New York.

    Google Scholar 

  • Poincarée, H. [1901], Sur une forme nouvelle des éequations de la méecanique, C. R. Acad. Sci. Paris 132, 369–371.

    Google Scholar 

  • Schwinger, J. [1951], On gauge invariance and vacuum polarization, Phys. Rev. 82, 664–679.

    Article  MATH  MathSciNet  Google Scholar 

  • Schwinger, J. [1959], Field theory commutators, Phys. Rev. Lett. 3, 296–297.

    Article  Google Scholar 

  • Serrin, J. [1959], in Mathematical Principles of Classical Fluid Mechanics, vol. VIII/1 of Encyclopedia of Physics, (S. Flüugge, ed.), Springer-Verlag, Berlin, Sections 14–15, pp. 125–263.

    Google Scholar 

  • Stern, A. [1999], Duality for coset models, Nuc. Phys. B 557, 459–479.

    Article  MATH  Google Scholar 

  • Trebin, H. R. [1982], The topology of non-uniform media in condensed matter physics, Adv. in Physics 31, 195–254.

    Article  MathSciNet  Google Scholar 

  • Tsurumaru, T. and I. Tsutsui [1999], On topological terms in the O(3) nonlinear sigma model, Phys. Lett. B 460, 94–102.

    Article  MathSciNet  MATH  Google Scholar 

  • Volovick, G. E. [1992], Exotic Properties of Superfluid 3 He, World-Scientific, Singapore.

    Google Scholar 

  • Volovick, G. E. and T. Vachaspati [1996], Aspects of 3He and the standard electroweak model, Internat. J. Mod. Phys. B 10, 471–521.

    Article  Google Scholar 

  • Volovik, G. E. and V. S. Dotsenko [1980], Hydrodynamics of defects in condensed media in the concrete cases of vortices in rotating Helium-II and of disclinations in planar magnetic substances, Sov. Phys. JETP, 58 65–80. [Zh. Eksp. Teor. Fiz. 78, 132–148.]

    Google Scholar 

  • Weatherburn, C. E. [1974], Differential Geometry in Three Dimensions, vol. 1, Cambridge University Press.

    Google Scholar 

  • Weinstein, A. [1996], Lagrangian mechanics and groupoids, Fields Inst. Commun. 7, 207–231.

    MATH  MathSciNet  Google Scholar 

  • Yabu, H. and H. Kuratsuji [1999], Nonlinear sigma model Lagrangian for super-fluid 3He-A(B), J. Phys. A 32, 7367–7374.

    Article  MathSciNet  MATH  Google Scholar 

  • Zakharov, V. E. and E. A. Kusnetsov [1997], Hamiltonian formalism for nonlinear waves, Usp. Fiz. Nauk 167, 1137–1167.

    Article  Google Scholar 

  • Zakharov, V. E., S. L. Musher and A. M. Rubenchik [1985], Hamiltonian approach to the description of nonlinear plasma phenomena, Phys. Rep. 129, 285–366.

    Article  MathSciNet  Google Scholar 

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Authors
  1. Darryl D. Holm
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Editors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, 90089-1191, USA

    Paul Newton

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089-1191, USA

    Paul Newton

  3. Department of Applied and Computational Mathematics Engineering Quadrangle, Princeton University, Princeton, NJ, 08544-1000, USA

    Philip Holmes

  4. Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, USA

    Alan Weinstein

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To Jerry Marsden on the occasion of his 60th birthday

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Holm, D.D. (2002). Euler-Poincaré Dynamics of Perfect Complex Fluids. In: Newton, P., Holmes, P., Weinstein, A. (eds) Geometry, Mechanics, and Dynamics. Springer, New York, NY. https://doi.org/10.1007/0-387-21791-6_4

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  • DOI: https://doi.org/10.1007/0-387-21791-6_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-95518-6

  • Online ISBN: 978-0-387-21791-8

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