Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: V. I. Arnold and B. A. Khesin
Title: Topological methods in hydrodynamics
Additional book information: Springer, New York, 1998, xv + 374 pp., ISBN 0-387-94947-X, $59.95$

References [Enhancements On Off] (What's this?)

  • V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 202082
  • V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
  • Yann Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2 (1989), no. 2, 225–255. MR 969419, DOI 10.1090/S0894-0347-1989-0969419-8
  • Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI 10.1103/PhysRevLett.71.1661
    • 5.
    A. CONSTANTIN AND J. ESCHER, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181, (1998), 229-243. CMP 99:07
  • Ronald J. DiPerna and Andrew J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. MR 877643
  • David G. Ebin and Jerrold Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102–163. MR 271984, DOI 10.2307/1970699
  • B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47–66. MR 636470, DOI 10.1016/0167-2789(81)90004-X
  • Michael H. Freedman and Zheng-Xu He, Divergence-free fields: energy and asymptotic crossing number, Ann. of Math. (2) 134 (1991), no. 1, 189–229. MR 1114611, DOI 10.2307/2944336
  • Darryl D. Holm and Boris A. Kupershmidt, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Phys. D 6 (1983), no. 3, 347–363. MR 709394, DOI 10.1016/0167-2789(83)90017-9
  • Darryl D. Holm, Jerrold E. Marsden, and Tudor S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), no. 1, 1–81. MR 1627802, DOI 10.1006/aima.1998.1721
  • Darryl D. Holm, Jerrold E. Marsden, Tudor Ratiu, and Alan Weinstein, Nonlinear stability of fluid and plasma equilibria, Phys. Rep. 123 (1985), no. 1-2, 116. MR 794110, DOI 10.1016/0370-1573(85)90028-6
  • Jerrold E. Marsden, A group theoretic approach to the equations of plasma physics, Canad. Math. Bull. 25 (1982), no. 2, 129–142. MR 663606, DOI 10.4153/CMB-1982-019-9
  • Jerrold E. Marsden and Tudor S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1994. A basic exposition of classical mechanical systems. MR 1304682, DOI 10.1007/978-1-4612-2682-6
    • 15.
    J.E. MARSDEN, T.S. RATIU, AND S. SHKOLLER, A nonlinear analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal., to appear.
  • Jerrold E. Marsden, Tudor S. Raţiu, and Alan Weinstein, Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984), no. 1, 147–177. MR 719663, DOI 10.1090/S0002-9947-1984-0719663-1
  • Jerrold E. Marsden and Jürgen Scheurle, The reduced Euler-Lagrange equations, Dynamics and control of mechanical systems (Waterloo, ON, 1992) Fields Inst. Commun., vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 139–164. MR 1232914, DOI 10.1007/bf00914351
  • Jerrold Marsden and Alan Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Phys. D 7 (1983), no. 1-3, 305–323. Order in chaos (Los Alamos, N.M., 1982). MR 719058, DOI 10.1016/0167-2789(83)90134-3
  • Jerrold E. Marsden and Alan Weinstein, The Hamiltonian structure of the Maxwell-Vlasov equations, Phys. D 4 (1981/82), no. 3, 394–406. MR 657741, DOI 10.1016/0167-2789(82)90043-4
  • Gerard Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys. 24 (1998), no. 3, 203–208. MR 1491553, DOI 10.1016/S0393-0440(97)00010-7
    • 21.
    S. SHKOLLER, Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics, J. Funct. Anal., 160, (1998), 337-365. CMP 99:06
    22.
    S. SHKOLLER, The geometry and analysis of non-Newtonian fluids and vortex methods, (1999), E-print math.AP/9908109.
  • A. I. Shnirel′man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.) 128(170) (1985), no. 1, 82–109, 144 (Russian). MR 805697
    • Review Information:

    Reviewer: Steve Shkoller
    Affiliation: University of California, Davis
    Email: shkoller@math.ucdavis.edu
    Journal: Bull. Amer. Math. Soc. 37 (2000), 175-181
    DOI: https://doi.org/10.1090/S0273-0979-99-00855-1
    Published electronically: December 21, 1999
    Review copyright: © Copyright 2000 American Mathematical Society