Mathematics

Dynamical Systems & Differential Equations
 Vladimir I. Arnold  Collected Works  Representations of Functions, Celestial Mechanics, and
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Collected Works of one of the most outstanding mathematicians of all times.
Vladimir Igorevich Arnold is one of the most influential mathematicians of our time. V. I. Arnold launched several mathematical domains (such as modern geometric mechanics, symplectic topology, and topological fluid dynamics) and contributed, in a fundamental way, to the foundations and methods in many subjects, from ordinary differential equations and celestial mechanics to singularity theory and real algebraic geometry. Even a quick look at a partial list of notions named after Arnold already gives an overview of the variety of such theories and domains: KAM (Kolmogorov–Arnold–Moser) theory, The Arnold conjectures in symplectic topology, The Hilbert–Arnold problem for the number of zeros of abelian integrals, Arnold’s inequality, comparison, and complexification method in real algebraic geometry, Arnold–Kolmogorov solution of Hilbert’s 13th problem, Arnold’s spectral sequence in singularity theory, Arnold diffusion, The Euler–Poincaré–Arnold equations for geodesics on Lie groups, Arnold’s stability criterion in hydrodynamics, ABC (Arnold–Beltrami–Childress) ?ows in ?uid dynamics, The Arnold–Korkina dynamo, Arnold’s cat map, The Arnold–Liouville theorem in integrable systems, Arnold’s continued fractions, Arnold’s interpretation of the Maslov index, Arnold’s relation in cohomology of braid groups, Arnold tongues in bifurcation theory, The Jordan–Arnold normal forms for families of matrices, The Arnold invariants of plane curves. Arnold wrote some 700 papers, and many books, including 10 university textbooks. He is known for his lucid writing style, which combines mathematical rigour with physical and geometric intuition. Arnold’s books on Ordinarydifferentialequations and Mathematical methodsofclassicalmechanics became mathematical bestsellers and integral parts of the mathematical education of students throughout the world.
Content Level »Research
Keywords »Celestial Mechanics  KAM theory  Representations of Functions  celestial mechanic  ordinary differential equation
On the representation of functions of two variables in the form ?[?(x) + ?(y)]. On functions of three variables. The mathematics workshop for schools at Moscow State University. The school mathematics circle at Moscow State University: harmonic functions. On the representation of functions of several variables as a superposition of functions of a smaller number of variables. Representation of continuous functions of three variables by the superposition of continuous functions of two variables. Some questions of approximation and representation of functions. Kolmogorov seminar on selected questions of analysis. On analytic maps of the circle onto itself. Small denominators. I. Mapping of the circumference onto itself. The stability of the equilibrium position of a Hamiltonian system of ordinary differential equations in the general elliptic case. Generation of almost periodic motion from a family of periodic motions. Some remarks on flows of line elements and frames. A test for nomographic representability using Decartes’ rectilinear abacus. Remarks on winding numbers. On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian. Small perturbations of the automorphisms of the torus. The classical theory of perturbations and the problem of stability of planetary systems. Letter to the editor. Dynamical systems and group representations at the Stockholm Mathematics Congress. Proof of a theorem of A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian. Small denominators and stability problems in classical and celestial mechanics. Small denominators and problems of stability of motion in classical and celestial mechanics. Uniform distribution of points on a sphere and some ergodic properties of solutions of linear ordinary differential equations in a complex region. On a theorem of Liouville concerning integrable problems of dynamics. Instability of dynamical systems with several degrees of freedom. On the instability of dynamical systems with several degrees of freedom. Errata to V.I. Arnol’d’s paper: “Small denominators. I.”. Small denominators and the problem of stability in classical and celestial mechanics. Stability and instability in classical mechanics. 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. On a topological property of globally canonical maps in classical mechanics.