Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 030, 22 pages      arXiv:0803.1678      http://dx.doi.org/10.3842/SIGMA.2008.030
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Geodesic Equations on Diffeomorphism Groups

Cornelia Vizman
Department of Mathematics, West University of Timisoara, Romania

Received November 13, 2007, in final form March 01, 2008; Published online March 11, 2008

Abstract
We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L2 or H1 metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.

Key words: Euler's equation; diffeomorphism group; group extension; geodesic equation.

pdf (365 kb)   ps (232 kb)   tex (37 kb)

References

  1. Alekseevsky A., Michor P.W., Ruppert W., Extensions of super Lie algebras, J. Lie Theory 15 (2005), 125-134, math.QA/0101190.
  2. Arnold V.I., Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), 319-361.
  3. Arnold V.I., Khesin B.A., Topological methods in hydrodynamics, Springer, Berlin, 1998.
  4. Bao D., Ratiu T., On the geometrical origin of a degenerate Monge-Ampère equation, Proc. Sympos. Pure Math. 54 (1993), 55-68.
  5. Billig Y., Magnetic hydrodynamics with asymmetric stress tensor, J. Math. Phys. 46 (2005), 043101, 13 pages, math-ph/0401052.
  6. Brenier Y., Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math. 52 (1999), 411-452.
  7. Burgers J., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171-199.
  8. Camassa R., Holm D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), 1661-1664, patt-sol/9305002.
  9. Constantin A., Kappeler T., Kolev B., Topalov P., On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007), 155-180.
  10. Ebin D., Marsden J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102-163.
  11. Eliashberg Ya., Ratiu T., The diameter of the symplectomorphism group is infinite, Invent. Math. 103 (1991), 327-340.
  12. Fuks D.B., Cohomology of infinite-dimensional lie algebras, Contemp. Sov. Math., Consultants Bureau, New York, 1986.
  13. Gay-Balmaz F., Ratiu T., The Lie-Poisson structure of the LAE-α equation, Dyn. Partial Differ. Equ. 2 (2005), 25-57, math.DG/0504381.
  14. Gay-Balmaz F., Ratiu T., Euler-Poincaré and Lie-Poisson formulations of Euler-Yang-Mills equations, Preprint, 2006.
  15. Gay-Balmaz F., Ratiu T., Well posedness of higher dimensional Camassa-Holm equations on manifolds, Preprint, 2007.
  16. Haller S., Teichmann J., Vizman C., Totally geodesic subgroups of diffeomorphisms, J. Geom. Phys. 42 (2002), 342-354, math.DG/0103220.
  17. Hattori Y., Ideal magnetohydrodynamics and passive scalar motion as geodesics on semidirect product groups, J. Phys. A: Math. Gen. 27 (1994), L21-L25.
  18. Hirani A.N., Marsden J.E., Arvo J., Averaged template matching equations, Lecture Notes Computer Science, Vol. 2134, Proc. Third Int. Workshop EMMCVPR, 2001, Springer, Berlin-Heidelberg, 528-543.
  19. Holm D., Marsden J., Ratiu T., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1997), 1-81, chao-dyn/9801015.
  20. Holm D., Marsden J., Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation, in Festschrift for Alan Weinstein, Birkhäuser, Boston, 2003, 203-235, nlin.CD/0312048.
  21. Holm D., Zeitlin V., Hamilton's principle for quasigeostrophic motion, Phys. Fluids 10 (1998), 800-806, chao-dyn/9801018.
  22. Hunter J.K., Saxton R., Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521.
  23. Ismagilov R.S., Representations of infinite-dimensional groups, AMS Translations of Mathematical Monographs, Vol. 152, American Mathematical Society, Providence, RI, 1996.
  24. Ismagilov R.S., Inductive limits of area-preserving diffeomorphism groups, Funct. Anal. Appl. 37 (2003), 191-202.
  25. Khesin B., Misiolek G., Euler equations on homogeneous spaces and Virasoro orbits, Adv. Math. 176 (2003), 116-144, math.SG/0210397.
  26. Khesin B., Misiolek G., Asymptotic directions, Monge-Ampère equations and the geometry of diffeomorphism groups, J. Math. Fluid. Mech. 7 (2005), S365-S375, math.DG/0504556.
  27. Kirillov A.A., The orbit method. II. Infinite dimensional Lie groups and Lie algebras, Contemp. Math. 145 (1993), 33-63.
  28. Kriegl A., Michor P.W., The convenient setting of global analysis, Mathematical Surveys and Monographs, Vol. 53, Amer. Math. Soc., Providence, RI, 1997.
  29. Kouranbaeva S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys. 40 (1999), 857-868, math-ph/9807021.
  30. Marsden J., Ratiu T., Introduction to mechanics and symmetry, 2nd ed., Springer, 1999.
  31. Marsden J., Ratiu T., Shkoller S., The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal. 10 (2000), 582-599, math.AP/9908103.
  32. Marsden J.E., Ratiu T., Weinstein A., Semidirect product and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984), 147-177.
  33. Michor P.W., Mumford D., Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Doc. Math. 10 (2005), 217-245, math.DG/0409303.
  34. Michor P.W., Ratiu T., On the geometry of the Virasoro-Bott group, J. Lie Theory 8 (1998), 293-309, math.DG/9801115.
  35. Milnor J., Remarks on infinite-dimensional Lie groups, in Proc. Summer School on Quantum Gravity, Editor B. DeWitt, North Holland, Amsterdam, 1984, 1007-1057.
  36. Misiolek G., Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms, Indiana Univ. Math. J. 42 (1993), 215-235.
  37. Misiolek G., Conjugate points in the Bott-Virasoro group and the KdV equation, Proc. Amer. Math. Soc. 125 (1997), 935-940.
  38. Misiolek G., A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), 203-208.
  39. Misiolek G., Classical solutions of the periodic Camassa-Holm equation, Geom. Funct. Anal. 12 (2002), 1080-1104.
  40. Misiolek G., Conjugate points in SDiff(T2), Proc. Amer. Math. Soc. 124 (1996), 977-982.
  41. Moreau J.J., Une méthode de ''cinématique fonctionnelle'' en hydrodynamique, C. R. Acad. Sci. Paris 249 (1959), 2156-2158.
  42. Nakamura F., Hattori Y., Kambe T., Geodesics and curvature of a group of diffeomorphisms and motion of an ideal fluid, J. Phys. A: Math. Gen. 25 (1992), L45-L50.
  43. Neeb K.-H., Abelian extensions of infinite-dimensional Lie groups, Travaux Math. XV (2004), 69-194, math.GR/0402303.
  44. Ovsienko V.Y., Khesin B.A., Korteweg-de Vries superequations as an Euler equation, Funct. Anal. Appl. 21 (1987), 329-331.
  45. Ovsienko V., Roger C., Generalizations of Virasoro group and Virasoro algebra through extensions by modules of tensor densities on S1, Indag. Math. (N.S.) 9 (1998), 277-288.
  46. Pekarsky S., Shkoller S., On the stability of periodic 2D Euler-α flows, math.AP/0007050.
  47. Poincaré H., Sur une nouvelle forme des équations de la méchanique, C. R. Acad. Sci. 132 (1901), 369-371.
  48. Preston S.C., For ideal fluids, Eulerian and Lagrangian instabilities are equivalent, Geom. Funct. Anal. 14 (2004), 1044-1062.
  49. Roger C., Extensions centrales d'algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et généralisations, Rep. Math. Phys. 35 (1995), 225-266.
  50. Shkoller S., Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics, J. Funct. Anal. 160 (1998), 337-365, math.AP/9807078.
  51. Shkoller S., Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom. 55 (2000), 145-191.
  52. Shnirel'man A.I., The geometry of the group of diffeomorphisms and the dynamics of an incompressible fluid, Math. Sb. 56 (1987), 79-105.
  53. Shnirel'man A.I., Generalized fluid flows, their approximation and applications, Geom. Funct. Anal. 4 (1994), 586-620.
  54. Vishik S.M., Dolzhanskii F.V., Analogs of the Euler-Lagrange equations and magnetohydrodynamics equations related to Lie groups, Sov. Math. Doklady 19 (1978), 149-153.
  55. Vizman C., Curvature and geodesics on diffeomorphism groups, in Proceedings of the Fourth International Workshop on Differential Geometry, Brasov, Romania, 1999, 298-305.
  56. Vizman C., Geodesics and curvature of semidirect product groups, Rend. Circ. Mat. Palermo (2) Suppl. 66 (2001), 199-206, math.DG/0103141.
  57. Vizman C., Geodesics on extensions of Lie groups and stability: the superconductivity equation, Phys. Lett. A 284 (2001), 23-30, math.DG/0103140.
  58. Vizman C., Central extensions of semidirect products and geodesic equations, Phys. Lett. A 330 (2004), 460-469.
  59. Vizman C., Central extensions of the Lie algebra of symplectic vector fields, J. Lie Theory 16 (2005), 297-309.
  60. Vizman C., Quasigeostrophic motion, stream functions and cocycles, J. Nonlinear Math. Phys., to appear.
  61. Zeitlin V., Vorticity and waves: geometry of phase-space and the problem of normal variables, Phys. Lett. A 164 (1992), 177-183.
  62. Zeitlin V., Pasmanter R.A., On the differential geometric approach to geophysical flows, Phys. Lett. A 189 (1994), 59-63.
  63. Zeitlin V., Kambe T., Two-dimensional ideal magnetohydrodynamics and differential geometry, J. Phys. A: Math. Gen. 26 (1993), 5025-5031.

Previous article   Next article   Contents of Volume 4 (2008)