The geometric structure of complex fluids

Advances in Applied Mathematics

Volumn 42, Issue 2, 2009, Pages 176-275

  Gay Balmaz, François ^a   Ratiu, Tudor S ^a  

^a EPFL   (Switzerland)

Author keywords

Affine Euler Poincar equations; Affine Lie Poisson equations; Complex fluids; Diffeomorphism group; Hall magnetohydrodynamics; Liquid crystals; Microfluids; Poisson brackets; Spin glasses; Superfluid dynamics; Yang Mills magnetohydrodynamics

Indexed keywords

CRYSTALS; DYNAMICS; EULER EQUATIONS; FASTENERS; FLUID DYNAMICS; FLUIDS; GLASS; HYDRODYNAMICS; LAGRANGE MULTIPLIERS; LIGHT SOURCES; LIQUID CRYSTALS; LIQUIDS; MAGNETIC FIELD EFFECTS; POISSON DISTRIBUTION; POISSON EQUATION; PROGRAMMING THEORY; SPIN DYNAMICS;

AFFINE LIE-POISSON EQUATIONS; COMPLEX FLUIDS; DIFFEOMORPHISM GROUP; HALL MAGNETOHYDRODYNAMICS; MICROFLUIDS; POISSON BRACKETS; SPIN GLASSES; SUPERFLUID DYNAMICS; YANG-MILLS MAGNETOHYDRODYNAMICS;

MAGNETOHYDRODYNAMICS;

EID: 58549114848     PISSN: 01968858     EISSN: 10902074     Source Type: Journal    
DOI: 10.1016/j.aam.2008.06.002     Document Type: Article

References (30)

1. Abraham R., Marsden J.E., and Ratiu T.S. Manifolds, Tensor Analysis, and Applications. Appl. Math. Sci. second enlarged ed. vol. 75 (1989), Springer-Verlag
2. Bloch A.M., Krishnaprasad P.S., Marsden J.E., and Ratiu T.S. The Euler-Poincaré equations and double bracket dissipation. Comm. Math. Phys. 175 (1996) 1-42
3. Cendra H., Marsden J.E., and Ratiu T.S. Cocycles, compatibility, and Poisson brackets for complex fluids. In: Capriz G., and Mariano P.M. (Eds). Advances in Multifield Theories of Continua with Substructure. Model. Simul. Sci. Eng. Technol. (2003), Birkhäuser, Boston 51-73
4. Chandrasekar S. Liquid Crystals. second ed. (1992), Cambridge Univ. Press, Cambridge
5. de Gennes P.G., and Prost J. The Physics of Liquid Crystals. second ed. (1993), Oxford Univ. Press, Oxford
6. Dzyaloshinskiǐ I.E. Macroscopic description of spin glasses. Modern Trends in the Theory of Condensed Matter (Proc. Sixteenth Karpacz Winter School Theoret. Phys.). Karpacz, 1979. Lecture Notes in Phys. vol. 115 (1980) 204-224
7. Eringen A.C. Microcontinuum Field Theories I: Foundations and Solids (1999), Springer-Verlag Microcontinuum Field Theories II: Fluent Media
8. Eringen A.C. Microcontinuum Field Theories I: Foundations and Solids (2001), Springer-Verlag
9. Gay-Balmaz F., and Ratiu T.S. Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids. J. Symplectic Geom. 6 (2008) 189-237
10. F. Gay-Balmaz, T.S. Ratiu, Affine Lie-Poisson reduction, Yang-Mills magnetohydrodynamics, and superfluids, J. Phys. A 41 (2008), in press
11. Gibbons J., Holm D.D., and Kupershmidt B.A. The Hamiltonian structure of classical chromohydrodynamics. Phys. D 6 (1983) 179-194
12. Holm D.D. Hall magnetohydrodynamics: Conservation laws and Lyapunov stability. Phys. Fluids 30 (1987) 1310-1322
13. Holm D.D. Introduction to HVBK Dynamics. Lecture Notes in Phys. vol. 571 (2001), Springer-Verlag pp. 114-130
14. Holm D.D. Euler-Poincaré dynamics of perfect complex fluids. In: Holmes P., Newton P., and Weinstein A. (Eds). Geometry, Dynamics and Mechanics: 60th Birthday Volume for J.E. Marsden (2002), Springer-Verlag 113-167
15. Holm D.D., and Kupershmidt B.A. Yang-Mills magnetohydrodynamics: Nonrelativistic theory. Phys. Rev. D 30 (1984) 2557-2560
16. Holm D.D., and Kupershmidt B.A. Superfluid plasmas: Multivelocity nonlinear hydrodynamics of superfluid solutions with charged condensates coupled electromagnetically. Phys. Rev. A 36 (1987) 3947-3956
17. Holm D.D., and Kupershmidt B.A. The analogy between spin glasses and Yang-Mills fluids. J. Math. Phys. 29 (1988) 21-30
18. Holm D.D., Marsden J.E., and Ratiu T.S. The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137 (1998) 1-81
19. Isaev A.A., Kovalevskii M.Yu., and Peletminskii S.V. On dynamics of various magnetically ordered structures. Phys. Met. Metallogr. 77 4 (1994) 342-347
20. Ivanchenko E.A. Backward electromagnetic waves in a magnetically disordered dielectric. Low. Temp. Phys. 26 6 (2000) 422-424
21. Knapp A.W. Lie Groups. Beyond and Introduction. Progr. Math. vol. 140 (1996), Birkhäuser, Boston
22. Lhuillier D., and Rey A.D. Nematic liquid crystals and ordered micropolar fluids. J. Non-Newtonian Fluid Mech. 120 (2004) 169-174
23. Marsden J.E., Misiołek G., Ortega J.-P., Perlmutter M., and Ratiu T.S. Hamiltonian Reduction by Stages. Lecture Notes in Math. vol. 1913 (2007), Springer-Verlag
24. Marsden J.E., and Ratiu T.S. Introduction to Mechanics and Symmetry. Texts Appl. Math. vol. 17 (1994), Springer-Verlag second ed. 1999, second printing 2003
25. Marsden J.E., Ratiu T.S., and Weinstein A. Semidirect product and reduction in mechanics. Trans. Amer. Math. Soc. 281 (1984) 147-177
26. Morrison P.J., and Greene J.M. Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45 (1980) 790-794 Errata
27. Morrison P.J., and Greene J.M. Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 48 (1982) 569
28. Skyrme T.H.R. A nonlinear field theory. Proc. R. Soc. Lond. Ser. A 260 (1961) 127-138
29. Volovik G.E., and Dotsenko V.S. Hydrodynamics of defects in condensed media, using as examples vortices in rotating He II and disclinations in a planar magnet. Soviet Phys. JETP 78 1 (1980) 132-148 (in Russian)
30. Virga E.G. Variational Theories for Liquid Crystals (1994), Chapman & Hall