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,
$59.95$
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
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
- 1.
- V.I. ARNOLD, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluids parfaits, Ann. Inst. Grenoble, 16, (1966), 319-361. MR 0202082
- 2.
- V.I. ARNOLD, Mathematical methods of classical mechanics, Springer-Verlag, GTM 60, 1978. MR 0474391
- 3.
- Y. 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 0969419; The dual least action problem for an ideal, incompressible fluid, Arch. Rational Mech. Anal., 122, (1993), no. 4, 323-351, MR 1217592; Minimal geodesics on groups of volume-preserving maps and generalized solutions to the Euler equations, Comm. Pure Appl. Math., 52, (1999), 411-452. CMP 99:06
- 4.
- R. CAMASSA AND D.D. HOLM An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, (1993), 1661-1664. MR 1234453
- 5.
- A. CONSTANTIN AND J. ESCHER, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181, (1998), 229-243. CMP 99:07
- 6.
- R. DIPERNA AND A. MAJDA, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108, (1987), 667-689. MR 0877643
- 7.
- D. EBIN AND J. MARSDEN, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92, (1970), 102-163. MR 0271984
- 8.
- B. FUCHSSTEINER AND A.S. FOKAS, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. MR 0636470
- 9.
- M.H. FREEDMAN AND Z.-X., HE, Divergence-free fields: energy and asymptotic crossing number, Annals of Math., 134, (1991), no. 1, 189-229. MR 1114611
- 10.
- D.D. HOLM, B.A. KUPERSHMIDT, Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity, Phys. D 6 (1983), no. 3, 347-363. MR 0709394
- 11.
- D.D. HOLM, J.E. MARSDEN, AND T.S. RATIU, Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math. 137, (1998), 1-81. MR 1627802
- 12.
- D.D. HOLM, J.E MARSDEN, T. RATIU, AND A. WEINSTEIN, Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123, (1985), 1-116. MR 0794110
- 13.
- J.E. MARSDEN, A group theoretic approach to the equations of plasma physics, Canad. Math. Bull., 25, (1982), no. 2, 129-142. MR 0663606
- 14.
- J.E. MARSDEN AND T.S. RATIU, Introduction to Mechanics and Symmetry, Springer-Verlag, Second Edition, 1999. MR 1304682
- 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.
- 16.
- MARSDEN, J.E., T. RATIU, AND A. WEINSTEIN, Semidirect products and reduction in mechanics, Trans. Am. Math. Soc., 281, (1984), 147-177. MR 0719663
- 17.
- J.E. MARSDEN AND J. SCHEURLE, The reduced Euler-Lagrange equations, Dynamics and control of mechanical systems (Waterloo, ON, 1992), 139-164, Fields Inst. Commun., 1, Amer. Math. Soc., Providence, RI, 1993. MR 1232914
- 18.
- MARSDEN, J.E. AND A. WEINSTEIN, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D, 7, (1983), 305-323. MR 0719058
- 19.
- MARSDEN, J.E. AND A. WEINSTEIN, The Hamiltonian structure of the Maxwell-Vlasov equations, Phys. D, 4 (1982), no. 3, 394-406. MR 0657741
- 20.
- G. MISIOEK, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24, (1998), 203-208. MR 1491553
- 21.
- S. SHKOLLER, Geometry and curvature of diffeomorphism groups with
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.
- 23.
- A.I. SHNIRELMAN, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170), (1985), 82-109, 144, MR 0805697; Attainable diffeomorphisms, Geom. Funct. Anal., 3, (1993), 279-294, MR 1215782; Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4, (1994), 586-620. MR 1296569
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