Mathematical Structures in Continuous Dynamical Systems
Poisson Systems and Complete Integrability with Applications from Fluid Dynamics
Seiten
1995
Elsevier Science Ltd (Verlag)
978-0-444-82151-5 (ISBN)
Elsevier Science Ltd (Verlag)
978-0-444-82151-5 (ISBN)
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This work addresses several aspects of continuous dynamical systems, all of which can be viewed as generalizations of methods from classical mechnics. Equations such as the Korteweg-de Vries, non-linear Schrodinger, Sine-Gordon and Boussinesq equations are treated in detail.
In mathematical physics various phenomena from nature are described at each instant with an infinite-dimensional state variable (a function of spatial variables, in general), and basic laws of physics describe the evolution. One important area of research, both for physical reasons and for the advancement of mathematical methods is fluid dynamics. Mathematically speaking, the state variable evolves according to a partial differential equation, an "evolution equation" describing the dynamical system. Dynamical systems for discrete (finite dimensional) systems, have been studied at length in classical mechanics, and new results and ideas such as chaos, are abundant. Although much more complicated than discrete systems, new developments for continuous systems (with spatial variations) are impressive. This book addresses several aspects, all of which can be viewed as generalizations of methods from classical mechanics. It explains in various ways how physical structures can be expected as a consequence of the underlying mathematical structure of the equation.
Complete integrability is one such mathematical structure, but systems with a less restrictive Poisson (or Hamiltonian) structure can also exhibit the same properties. Famous equations like the Korteweg - de Vries, nonlinear Schrodinger, Sine-Gordon, Boussinesq equations are treated in detail. The book is divided into two parts. Part I deals with (general) Poisson systems, mainly for problems from fluid dynamics. Wave equations and the equations for vortical flows are the prime examples. Part II provides an introduction to the mathematical theory of solitons.
In mathematical physics various phenomena from nature are described at each instant with an infinite-dimensional state variable (a function of spatial variables, in general), and basic laws of physics describe the evolution. One important area of research, both for physical reasons and for the advancement of mathematical methods is fluid dynamics. Mathematically speaking, the state variable evolves according to a partial differential equation, an "evolution equation" describing the dynamical system. Dynamical systems for discrete (finite dimensional) systems, have been studied at length in classical mechanics, and new results and ideas such as chaos, are abundant. Although much more complicated than discrete systems, new developments for continuous systems (with spatial variations) are impressive. This book addresses several aspects, all of which can be viewed as generalizations of methods from classical mechanics. It explains in various ways how physical structures can be expected as a consequence of the underlying mathematical structure of the equation.
Complete integrability is one such mathematical structure, but systems with a less restrictive Poisson (or Hamiltonian) structure can also exhibit the same properties. Famous equations like the Korteweg - de Vries, nonlinear Schrodinger, Sine-Gordon, Boussinesq equations are treated in detail. The book is divided into two parts. Part I deals with (general) Poisson systems, mainly for problems from fluid dynamics. Wave equations and the equations for vortical flows are the prime examples. Part II provides an introduction to the mathematical theory of solitons.
Part 1 Poisson structures in fluid dynamics: Poisson structures; surface waves; Eulerian fluid dynamics; consistent modelling; Poisson dynamics; coherent structures as relative equilibria; Poisson perturbation methods. Part 2 Mathematical introduction to the theory of solitons: solitons in physics and mathematics; A.K.N.S. systems and soliton equations; scattering, inverse scattering and solitons; backlund-transformations; the KdV-hierarchy as a hierarchy of Hamiltonian systems; prolongation structures.
| Reihe/Serie | Studies in Mathematical Physics S. ; v. 6 |
|---|---|
| Zusatzinfo | index |
| Verlagsort | Oxford |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Angewandte Mathematik |
| Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik | |
| ISBN-10 | 0-444-82151-1 / 0444821511 |
| ISBN-13 | 978-0-444-82151-5 / 9780444821515 |
| Zustand | Neuware |
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