Lectures on Topological Fluid Mechanics

Mitchell A. Berger, Louis H. Kauffman, Boris Khesin, H. Keith Moffatt, Renzo L. Ricca, De Witt Sumners (auth.), Renzo L. Ricca (eds.)3642008364, 9783642008368

Helmholtz’s seminal paper on vortex motion (1858) marks the beginning of what is now called topological fluid mechanics.After 150 years of work, the field has grown considerably. In the last several decades unexpected developments have given topological fluid mechanics new impetus, benefiting from the impressive progress in knot theory and geometric topology on the one hand, and in mathematical and computational fluid dynamics on the other.

This volume contains a wide-ranging collection of up-to-date, valuable research papers written by some of the most eminent experts in the field. Topics range from fundamental aspects of mathematical fluid mechanics, including topological vortex dynamics and magnetohydrodynamics, integrability issues, Hamiltonian structures and singularity formation, to DNA tangles and knotted DNAs in sedimentation. A substantial introductory chapter on knots and links, covering elements of modern braid theory and knot polynomials, as well as more advanced topics in knot classification, provides an invaluable addition to this material.

Table of contents :
Front Matter….Pages 1-12
Braids and Knots….Pages 1-73
Topological Quantities: Calculating Winding, Writhing, Linking, and Higher Order Invariants….Pages 75-97
Tangles, Rational Knots and DNA….Pages 99-138
The Group and Hamiltonian Descriptions of Hydrodynamical Systems….Pages 139-155
Singularities in Fluid Dynamics and their Resolution….Pages 157-166
Structural Complexity and Dynamical Systems….Pages 167-186
Random Knotting: Theorems, Simulations and Applications….Pages 187-217
Back Matter….Pages 219-231