Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom (eBook)

Vadim Kaloshin, Ke Zhang (Autoren)

eBook Download: PDF

2020
224 Seiten
Princeton University Press (Verlag)
978-0-691-20493-2 (ISBN)

Infos

The first complete proof of Arnold diffusion-one of the most important problems in dynamical systems and mathematical physicsArnold 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.

Produkt-Details

• Erscheint lt. Verlag: 3.11.2020
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 21 b/w illus.
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Schlagworte: action variables • Addition • Affine manifold • analytic function • Arnold’s paper • autonomous Hamiltonian system • average • Barrier function • Bifurcation Theory • Birkhoff (crater) • Boundary value problem • C0 • canonical form • Celestial mechanics • Characterization (mathematics) • Codimension • cohomology • Compact space • Configuration space • Connected component (graph theory) • conservation of action variables • Contradiction • coordinate system • corollary • Covering space • Curve • Degeneracy (mathematics) • Degrees of freedom (statistics) • Dense set • dichotomy • diffusion process • Discrete time and continuous time • Eigenvalues and Eigenvectors • equivalence relation • ergodic hypothesis • estimation • ETH Zurich • Euler–Lagrange equation • existential quantification • foliation • Generic property • geodesic • Graph property • hamiltonian mechanics • Hamiltonian System • Heteroclinic orbit • holonomy • Homoclinic connection • Homoclinic orbit • Homotopy • Initial Condition • instability of dynamical systems • Integrable Hamiltonian Systems • integrable system • Invariant manifold • invariant measure • KAM Theorem • Lagrangian (field theory) • Limit point • Linearization • linearly stable • Local coordinates • magnetic fields • motion of charged particles • N0 • negligible friction • non integrable • Open problem • Parameter • Perturb • Perturbation theory (quantum mechanics) • Phase space • probability measure • Pushforward • Regime • Semi-continuity • Separatrix (mathematics) • several degrees of freedom • Smoothness • stable manifold • stable solution • submanifold • subsequence • Subset • Suggestion • symplectic vector space • Tangent Space • Theorem • Three-body problem • Topological space • torus • Transversal (geometry) • Unit circle • Variational method (quantum mechanics)
• ISBN-10: 0-691-20493-4 / 0691204934
• ISBN-13: 978-0-691-20493-2 / 9780691204932