Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
Ke Zhang, Vadim Kaloshin
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Naturwissenschaften, Medizin, Informatik, Technik / Naturwissenschaften allgemein
The first complete proof of Arnold diffusion—one of the most important problems in dynamical systems and mathematical physics
Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. In this groundbreaking book, Vadim Kaloshin and Ke Zhang provide the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom).
This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. Kaloshin and Zhang follow Mather's strategy but emphasize a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, this book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.
Compact space, Analytic function, Homotopy, Degeneracy (mathematics), integrable Hamiltonian systems, Estimation, Hamiltonian mechanics, Phase space, Symplectic vector space, Perturbation theory (quantum mechanics), Hamiltonian system, autonomous Hamiltonian system, Characterization (mathematics), Heteroclinic orbit, Regime, Transversal (geometry), Stable manifold, Graph property, Open problem, Separatrix (mathematics), Covering space, Lagrangian (field theory), Curve, Smoothness, Cohomology, Canonical form, Integrable system, Local coordinates, Ergodic hypothesis, KAM theorem, Euler–Lagrange equation, linearly stable, Corollary, non integrable, Dense set, Codimension, Invariant manifold, Degrees of freedom (statistics), Torus, Contradiction, Birkhoff (crater), N0, Bifurcation theory, Parameter, Arnold’s paper, Discrete time and continuous time, Unit circle, C0, Tangent space, Foliation, Generic property, Homoclinic connection, conservation of action variables, Topological space, Affine manifold, Diffusion process, Connected component (graph theory), Pushforward, celestial mechanics, Equivalence relation, Suggestion, Holonomy, Linearization, several degrees of freedom, ETH Zurich, perturb, Coordinate system, Theorem, negligible friction, instability of dynamical systems, Subsequence, stable solution, Invariant measure, Semi-continuity, Three-body problem, Submanifold, Initial condition, Existential quantification, Variational method (quantum mechanics), Dichotomy, Barrier function, magnetic fields, Geodesic, action variables, Subset, Configuration space, Average, Boundary value problem, Eigenvalues and eigenvectors, Limit point, Addition, Homoclinic orbit, motion of charged particles, Probability measure