Vladimir I. Arnold - Collected Works: Hydrodynamics, Bifurcation Theory, and Algebraic Geometry 1965-1972

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Springer Science & Business Media, Dec 11, 2013 - Mathematics - 465 pages
Vladimir Arnold was one of the great mathematical scientists of our time. He is famous for both the breadth and the depth of his work. At the same time he is one of the most prolific and outstanding mathematical authors. This second volume of his Collected Works focuses on hydrodynamics, bifurcation theory, and algebraic geometry.
 

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Contents

1 A variational principle for threedimensional steady flows of an ideal fluid
iv
2 On the Riemann curvature of diffeomorphism groups
iv
3 Sur la topologie des écoulements stationnaires des fluides parfaits French
17
4 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid
21
5 On the topology of threedimensional steady flows of an ideal fluid
26
6 On an a priori estimate in the theory of hydrodynamical stability
30
7 On the differential geometry of infinitedimensional Lie groups and its applications to the hydrodynamics of perfect fluids
33
8 On a variational principle for the steady flows of perfect fluids and its applications to problems of nonlinear stability
70
25 Algebraic unsolvability of the problem of Lyapunov stability and the problem of topological classification of singular points
231
26 On the arrangement of ovals of real plane algebraic curves involutions of fourdimensional smooth manifolds and the arithmetic of integral quadra...
239
27 The topology of real algebraic curves works of IG Petrovsky and their development
250
28 On matrices depending on parameters
255
29 Lectures on bifurcations in versal families
270
30 Versal families and bifurcations of differential equations Russian
340
31 Remarks on the behaviour of a flow of a threedimensional i fluid in the presence of a small perturbation of the initial vector field
349
32 The asymptotic Hopf invariant and its applications
357

9 On a characteristic class arising in quantization conditions
85
10 A note on the Weierstrass preparation theorem
99
11 The stability problem and ergodic properties for classical dynamical systems
106
12 A remark on the ramification of hyperelliptic integrals as functions of parameters
115
13 Singularities of smooth mappings
119
14 Remarks on singularities of finite codimension in complex dynamical systems
162
15 Braids of algebraic functions and the cohomology of swallowtails
170
16 Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid
173
17 On the onedimensional cohomology of the Lie algebra of divergencefree vector fields and rotation numbers of dynamical systems
177
18 The cohomology ring of the colored braid group
181
19 On cohomology classes of algebraic functions invariant under Tschirnhausen transformations
187
20 Trivial problems
191
21 Local problems of analysis
192
22 Algebraic unsolvability of the problem of stability and the problem of topological classification of singular points of analytic systems of differenti...
197
23 On some topological invariants of algebraic functions
199
24 Topological invariants of algebraic functions II
223
33 A magnetic field in a moving conducting fluid with YaB Zeldovich AARuzmaikin and DD Sokolov
376
34 A magnetic field in a stationary flow with stretching in a Riemannian manifold with YaB Zeldovich AARuzmaikin and DD Sokolov
378
35 Stationary magnetic field in a periodic flow with YaB Zeldovich AARuzmaikin and DD Sokolov
383
36 Some remarks on the antidynamo theorem
386
37 Evolution of a magnetic field under the action of transfer and diffusion
396
38 The growth of a magnetic field in a threedimensional steady incompressible flow with EIKorkina
399
39 On evolution of a magnetic field under the action of drift and diffusion Russian
404
40 Exponential scattering of trajectories and its hydrodynamical applications
419
41 Kolmogorovs hydrodynamic attractors
429
42 Topological methods in hydrodynamics with BA Khesin
433
43 Translators preface to J Milnors book Morse Theory
455
Selected Works in Three Volumes Vol I New Methods of Celestial Mechanics
457
45 Comments on the paper On a geometric theorem by Henri Poincaré
462
Acknowledgements
464
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