Methods from contact and symplectic geometry can be used to solve highly non-trivial nonlinear partial and ordinary differential equations without resorting to approximate numerical methods or algebraic computing software. This book explains how it's done. It combines the clarity and accessibility of an advanced textbook with the completeness of an encyclopedia. The basic ideas that Lie and Cartan developed at the end of the nineteenth century to transform solving a differential equation into a problem in geometry or algebra are here reworked in a novel and modern way. Differential equations are considered as a part of contact and symplectic geometry, so that all the machinery of Hodge-deRham calculus can be applied. In this way a wide class of equations can be tackled, including quasi-linear equations and Monge-Ampere equations (which play an important role in modern theoretical physics and meteorology).
Rezensionen / Stimmen
'As a whole, (together with the many and very clearly worked out examples presented, which is one of the most important and highly appreciated merits of this book) the text is well written, very well organised and the exposition is very clear. So, I would allow myself to recommend it as a very useful stand-by introduction to the geometric view on linear and nonlinear differential equations.' Journal of Geometry and Symmetry in Physics
Reihe
Sprache
Verlagsort
Zielgruppe
Illustrationen
30 Tables, unspecified; 10 Halftones, unspecified; 48 Line drawings, unspecified
Maße
Höhe: 240 mm
Breite: 161 mm
Dicke: 35 mm
Gewicht
ISBN-13
978-0-521-82476-7 (9780521824767)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Alexei Kushner is a Professor and Dean of the Department of Mathematics and Computer Science, and a Senior Researcher at the Russian Academy of Sciences. Valentin Lychagin is a Professor at the Institute of Mathematics and Statistics, Tromso University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow. Vladimir Rubtsov is a Professor at the Departement de Mathematiques, Angers University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow.
Autor*in
Astrakhan State University, Russia
Universitetet i Tromso, Norway
Universite d'Angers, France
Introduction; Part I. Symmetries and Integrals: 1. Distributions; 2. Ordinary differential equations; 3. Model differential equations and Lie superposition principle; Part II. Symplectic Algebra: 4. Linear algebra of symplectic vector spaces; 5. Exterior algebra on symplectic vector spaces; 6. A Symplectic classification of exterior 2-forms in dimension 4; 7. Symplectic classification of exterior 2-forms; 8. Classification of exterior 3-forms on a 6-dimensional symplectic space; Part III. Monge-Ampere Equations: 9. Symplectic manifolds; 10. Contact manifolds; 11. Monge-Ampere equations; 12. Symmetries and contact transformations of Monge-Ampere equations; 13. Conservation laws; 14. Monge-Ampere equations on 2-dimensional manifolds and geometric structures; 15. Systems of first order partial differential equations on 2-dimensional manifolds; Part IV. Applications: 16. Non-linear acoustics; 17. Non-linear thermal conductivity; 18. Meteorology applications; Part V. Classification of Monge-Ampere Equations: 19. Classification of symplectic MAEs on 2-dimensional manifolds; 20. Classification of symplectic MAEs on 2-dimensional manifolds; 21. Contact classification of MAEs on 2-dimensional manifolds; 22. Symplectic classification of MAEs on 3-dimensional manifolds.
