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

Springer, 2014. — 464 p.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.A variational principle for three-dimensional steady flows of an ideal fluid // Published as "Variational Principle for three-dimensional steady-state flows of an ideal fluid" in J. Appl. Math. Mech. 29:5, 1002-1008, 1965. Translation of Prikl. Mat. Mekh, 29:5, 846-851, 1965.
On the Riemann curvature of diffeomorphism groups // Translation of C. R. Acad. Sc. Paris 260, 5668-5671, 1965. Translated by Denis Auroux.
Sur la topologie des ecoulements stationnaires des fluides parfaits (French) // С R. Acad. Sc. Paris 261, 17-20, 1965.
Conditions for non-linear stability of stationary plane curvilinear flows of an ideal fluid // Sov. Math. Dokl. 162, No. 5, 773-777, 1965. Translation of Dokl. Akad. Nauk SSSR, 162:5, 975-978, 1965.
On the topology of three-dimensional steady flows of an ideal fluid // J. Appl. Math. Mech. 30:1, 223-226, 1966. Translation of Prikl. Mat. Mekh. 30:1, 183-185, 1966.
On an a priori estimate in the theory of hydrodynamical stability // Am. Math. Soc. Transl. (2) 79, 267-269, 1969. Translation of lzv. Vyssh. Uchebn. Zaved. Mat. 5:54, 3-5, 1966.
On the differential geometry of infinite-dimensional Lie groups and its applications to the hydrodynamics of perfect fluids // Translation of Annales de L'Institut Fourier, Vol. 16, No. 1, 319-361, 1966. Translated by Alain Chenciner.
On a variational principle for the steady flows of perfect fluids and its applications to problems of non-linear stability // Translation of Journal de Mecanique, Vol. 5, No. 1, 29-43, 1966. Translated by Alain Chenciner.
On a characteristic class arising in quantization conditions // Published as "Characteristic class entering in quantization conditions " in Fund. Anal. Appl. 1, 1-13, 1967. Translation of Funkts. Anal. Prilozh. 1:1, 1-14, 1967.
A note on the Weierstrass preparation theorem // Published as "A note on Weierstrass' auxiliary theorem" in Fund. Anal. Appl. 1, 173-179, 1967. Translation of Funkts. Anal. Prilozh. 1:3, 1-8, 1967.
The stability problem and ergodic properties for classical dynamical systems // Am. Math. Soc. Transl. (2) 70, 5-11, 1969. Translation of Proc. Internat. Congr. Math., Moscow 1966, 387-392. 1968.
A remark on the ramification of hyperelliptic integrals as functions of parameters // Published as "Remark on the branching of hyperelliptic integrals as functions of the parameters" in Funct. Anal. Appl. 2, 187-189, 1968. Translation of Funkts. Anal. Prilozh. 2:3, 1-3, 1968. Translated by Victor Vassiliev.
Singularities of smooth mappings // Russ. Math. Surv. 23, 1-43, 1968. Translation of Usp. Mat. Nauk 23:1, 3-44, 1968.
Remarks on singularities of finite codimension in complex dynamical systems // Funct. Anal. Appl. 3, 1-5, 1969. Translation of Funkts. Anal. Prilozh. 3:1, 1-6, 1969. Translated by Victor Vassiliev.
Braids of algebraic functions and the cohomology of swallowtails // Translation of Usp. Mat. Nauk 23:4, 247-248, 1968. Translated by Gerald Gould.
Hamiltonian nature of the Euler equations in the dynamics of a rigid body and of an ideal fluid // Translation of Usp. Mat. Nauk 24:3, 225-226, 1969. Translated by Gerald Gould.
On the one-dimensional cohomology of the Lie algebra of divergence-free vector fields and rotation numbers of dynamical systems // Published as "One-dimensional cohomologies of the Lie algebras of nondivergent vector fields and rotation numbers of dynamic systems" in Funct. Anal. Appl. 3, 319-321, 1969. Translation of Funkts. Anal. Prilozh. 3:4, 77- 78, 1969. Translated by Victor Vassiliev.
The cohomology ring of the colored braid group // Math. Notes 5, 138-140, 1969. Translation of Mat. Zametki 5:2, 227-231, 1969. Translated by Victor Vassiliev.
On cohomology classes of algebraic functions invariant under Tschirnhausen transformations // Funct. Anal. Appl. 4, 74-75, 1970. Translation of Funkts. Anal. Prilozh. 4:1, 84-85, 1970. Translated by Victor Vassiliev.
Trivial problems // Translation of Proc. 5th Int. Conf. on Nonlinear Oscillations, Kiev 1969. Vol. 1, 630-631, Ukrain. Acad. Sciences, Kiev 1970. Translated by Gerald Gould.
Local problems of analysis // Moscow Univ. Math, Bull. 25 (1970), 77-80, 1970. Translation of Vestn. Mosk Univ. Ser. I, Mat. Mekh. 25:2, 52-56, 1970.
Algebraic unsolvability of the problem of stability and the problem of topological classification of singular points of analytic systems of differential equations (Russian) // Usp. Mat. Nauk 25:2, 265-266, 1970.
On some topological invariants of algebraic functions // Transact. Math. Moscow Soc. 21, 30-52, 1970. Translation of Tr. Mosk. Mat. Obsc. 27-46, 1970.
Topological invariants of algebraic functions II // Funct. Anal. Appl. 4, 91-98, 1970. Translation of Funkts. Anal. Prilozh. 4:2, 1-9, 1970.
Algebraic unsolvability of the problem of Lyapunov stability and the problem of topological classification of singular points of an analytic system of differential equations // Funct. Anal. Appl. 4, 173-180, 1970. Translation of Funkts. Anal. Prilozh, 4:3, 1-9, 1970.
On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms // Published as "Distribution of ovals of the real plane of algebraic curves, of involutions of four-dimensional smooth manifolds, and the arithmetic of integer-valued quadratic forms" in Funct. Anal. Appl. 5, 169-176, 1971. Translation of Funkts. Anal. Prilozh. 5:3, 1-9, 1971. Translated by Oleg Viro.
The topology of real algebraic curves (works of LG. Petrovsky and their development) // Translation of Usp. Mat. Nauk 28:5, 260-262, 1973. Translated by Oleg Viro.
On matrices depending on parameters // Russ. Math. Surv. 22, 29-43, 1971. Translation of Usp. Mat. Nauk 26:2, 101-114, 1971.
Lectures on bifurcations in versal families // Russ. Math. Surv. 27, 54-123, 1972. Translation of Usp. Mat. Nauk 27:5, 119-184, 1972.
Versal families and bifurcations of differential equations (Russian) // Izd. Inst. Akad. Nauk Ukrain. SSR, Kiev, 42-49, 1972.
Remarks on the behaviour of a flow of a three-dimensional ideal fluid in the presence of a small perturbation of the initial vector field // Published as "Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field" in J. Appl. Math. Mech. 36:2, 236-242, 1972. Translation of Prikl. Mat. Mekh, 36:2, 255-262, 1972.
The asymptotic Hopf invariant and its applications // Selecta Math. Sov. 5:4, 327-345, 1986. Translation of Proc. All-Union School in Diff. Eq. with Infinite Number of Variables and in Dyn. Syst. with Infinite Number of Degrees of Freedom, Dilhan 1973, 229-256, 1974.
A magnetic field in a moving conducting fluid (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov) // Translation of Usp. Mat. Nauk 36:5, 220-221, 1981. Translated by Gerald Gould.
A magnetic field in a stationary flow with stretching in a Riemannian manifold (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov) // Published as "A magnetic field in a stationary flow with stretching in Riemannian space" in Sov. Phys. JETP 54:6, 1083-1086, 1982. Translation of Zh. Eksp. Teor. Fiz. 81:6, 2052-2056, 1981.
Stationary magnetic field in a periodic flow (with Ya.B. Zeldovich, A.A.Ruzmaikin, and D.D. Sokolov) // Published as "Steady-state magnetic field in a periodic flow" in Sov. Phys. Dok. 27:10, 814-816, 1982. Translation of Dokl. Akad. Nauk SSSR, 266:6, 1357-1361, 1982.
Some remarks on the antidynamo theorem // Moscow Univ. Math. Bull. 37, 57-66, 1982. Translation of Vestn. Mosk. Univ. Ser. I, Mat. Mekh. 6, 50-57, 1982.
Evolution of a magnetic field under the action of transfer and diffusion // Translation of Usp. Mat. Nauk 38:2, 226-227, 1983. Translated by Gerald Gould.
The growth of a magnetic field in a three-dimensional steady incompressible flow (with E.I. Korkina) // Moscow Univ. Math. Bull. Ser. I, Math. Mech. 3, 50-54, 1983. Translation of Vestn. Mosk. Univ. Mat. 38:3, 43-46, 1983.
On evolution of a magnetic field under the action of drift and diffusion (Russian) // Some Problems in Modern Analysis, 8-21, MGU, Moscow 1984.
Exponential scattering of trajectories and its hydrodynamical applications // Translation of N.E. Kochin and the Development of Mechanics, 185-193, Nauka, Moscow 1984. Translated by Gerald Gould.
Kolmogorov's hydrodynamic attractors // Proc. Royal Soc. London A 434:1890, 19-22, 1991.
Topological methods in hydrodynamics (with B.A. Khesin) // Annu. Rev. Fluid Mech. 24, 145-166, 1992.
Translator's preface to J. Milnor's book "Morse Theory". - MIR, Moscow 1965. Translated by Gerald Gould.
Henri Poincare: Selected Works in Three Volumes. Vol. I New Methods of Celestial Mechanics - Preface. From the editorial board. Comments. - Nauka, Moscow 1971, 747-
752. Translated by Gerald Gould.
Comments on the paper "On a geometric theorem" by Henri Poincare // H. Poincare: Selected Works in Three Volumes. Vol. II. Klassiki Nauki, Nauka, Moscow 1972, 987-
989. Translated by Gerald Gould.