Arnold V.I. Collected Works. Vol. I: Representations of Functions, Celestial Mechanics, and KAM Theory 1957-1965

Springer, 2009. — 487 p.Vladimir Arnold is 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 first volume of his Collected Works focuses on representations of functions, celestial mechanics, and KAM theory.On the representation of functions of two variables in the form. // Uspekhi Mat. Nauk 12, No. 2, 119-121 (1957); translated by Gerald Gould.
On functions of three variables // Amer. Math. Soc. Transl. (2) 28 (1963), 51-
54. Translation of Dokl. Akad. Nauk SSSR 114:4 (1957), 679-681.
The mathematics workshop for schools at Moscow State University // Mat. Prosveshchenie 2, 241-245 (1957); translated by Gerald Gould.
The school mathematics circle at Moscow State University: harmonic functions (in Russian) // Mat. Prosveshchenie 3 (1958), 241-250.
On the representation of functions of several variables as a superposition of functions of a smaller number of variables // Mat. Prosveshchenie 3, 41-61 (1958); translated by Gerald Gould.
Representation of continuous functions of three variables by the superposition of continuous functions of two variables // Amer. Math. Soc. Transl. (2) 28 (1963), 61-
147. Translation of Mat. Sb. (n.S.) 48 (90): 1 (1959), 3-74 Corrections in Mat. Sb. (n.S.) 56 (98) :3 (1962), 392.
Some questions of approximation and representation of functions // Amer. Math. Soc. Transl. (2) 53 (1966), 192-
201. Translation of Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, New York, 1960, pp. 339-348.
Kolmogorov seminar on selected questions of analysis // Uspekhi Mat. Nauk 15, No. 1, 247-250 (1960); translated by Gerald Gould.
On analytic maps of the circle onto itself // Uspekhi Mat. Nauk 15, No. 2, 212-214 (1960) (Summary of reports announced to the Moscow Math. Soc); translated by Gerald Gould.
Small denominators. I. Mapping of the circumference onto itself // Amer. Math. Soc. Transl. (2) 46 (1965), 213-
284. Translation of lzv. Akad. Nauk SSSR Ser. Mat. 25:1 (1961). Corrections in lzv. Akad. Nauk SSSR Ser. Mat. 28:2 (1964), 479-480.
The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case // Soviet. Math. Dokl. 2 (1961). Translation of Dokl. Akad. Nauk SSSR 137:2 (1961), 255-257.
Generation of almost periodic motion from a family of periodic motions // Soviet. Math. Dokl. 2 (1961). Translation of Dokl. Akad. Nauk SSSR 138:1 (1961) 13-15.
Some remarks on flows of line elements and frames // Soviet. Math. Dokl. 2 (1961). Translation of Dokl. Akad. Nauk SSSR 138:2 (1961), 255-257.
A test for nomographic representability using Decartes' rectilinear abacus (in Russian) // Uspekhi Mat. Nauk 16:4 (1961).
Remarks on winding numbers (in Russian) // Sibirsk. Mat. Z. 2:6 (1961), 807-813.
On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian // Soviet. Math. Dokl. 3 (1962). Translation of Dokl. Akad. Nauk SSSR 142:4 (1962), 758-761.
Small perturbations of the automorphisms of the torus // Soviet. Math. Dokl. 3 (1962). Translation of Dokl. Akad. Nauk SSSR 144:2 (1962), 695-698, Corrections in Dokl. Akad. Nauk
SSSR, 1963, 150:5, 958.
The classical theory of perturbations and the problem of stability of planetary systems // Soviet. Math. Dokl. 3 (1962). Translation of Dokl. Akad. Nauk SSSR 145:3 (1962), 487-490.
Letter to the editor (in Russian) // Mat. Sb. (n.S.) 56 (98):3 (1962), 392.
Dynamical systems and group representations at the Stockholm Mathematics Congress (in Russian) // Uspekhi Mat. Nauk 18:2 (1963).
Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian // Russian Math. Surveys 18 (1963). Translation of Uspekhi Mat. Nauk 18:5 (1963), 13-40.
Small denominators and stability problems in classical and celestial mechanics (in Russian) // Problems of the motion of artificial satellites. Reports at the conference on general applied topics in theoretical astronomy (Moscow, 20-25 November 1961), USSR Academy of Sciences Publishing House, Moscow, 1963, pp. 7-17.
Small denominators and problems of stability of motion in classical and celestial mechanics // Russian Math Surveys 18 (1963). Translation of Uspekhi Mat. Nauk 18:6 (1963), 91-192, Corrections in Uspekhi Mat. Nauk 22:5 (1968), 276.
Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex region // Soviet Math. Dokl. 4 (1963). Translation of Dokl. Akad. Nauk SSSR 148:1 (1963), 9-12.
On a theorem of Liouville concerning integrable problems of dynamics // Amer. Math Soc. Transl. (2) 61(1967) 292-
296. Translation of Sibirsk. Mat. Z. 4:2 (1963).
Instability of dynamical systems with several degrees of freedom // Soviet Math. Dokl. 5 (1964). Translation of Dokl. Akad. Nauk SSSR 156:1 (1964), 9-12.
On the instability of dynamical systems with several degrees of freedom (in Russian) // Uspekhi Mat. Nauk 19:5 (1964), 181.
Errata to V.I. Arnol'd's paper: "Small denominators. I." // Izv. Akad. Nauk SSSR Ser. Mat. 28, 479-480 (1964); translated by Gerald Gould.
Small denominators and the problem of stability in classical and celestial mechanics (in Russian) // Proceedings of the Fourth АН-Union Mathematics Congress (Leningrad, 3-12 July 1961), vol. 2, Nauka, Leningrad, 1964, pp. 403-409.
Stability and instability in classical mechanics (in Russian) // Second Math. Summer School, Part 11 (Russian), Naukova Dumka, Kiev, 1965, pp. 85-119.
Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution // Soviet Math. Dokl. 6 (1965). Translation of Dokl. Akad. Nauk SSSR 161:1 (1965), 9-12.
On a topological property of globally canonical maps in classical mechanics // Translation of Sur une propriete topologique des applications globalement canoniques de la mecanique classique. С R. Acad. Sci. Paris 261:19 (1965) 3719-3722; translated by Alain Chenciner and Jaques Fejoz.