Contact Geometry and Nonlinear Differential Equations
Contact Geometry and Nonlinear Differential Equations
Valentin Lychagin , Universitetet i Tromsø, Norway
Vladimir Rubtsov , Université d'Angers, France
$194.00
USDMethods 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).
- Accessible and useful for both experts and non-specialists
- Many new ideas, first time available in book form
- Methods applicable to real world applications
Reviews & endorsements
"The book excels in clarity and accessibility on the one hand, and in completeness on the other hand."
Frans Cantrijn, Mathematical Reviews
Product details
April 2011Adobe eBook Reader
9780511889752
0 pages
0kg
58 b/w illus. 30 tables
This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
- 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-Ampère Equations:
- 9. Symplectic manifolds
- 10. Contact manifolds
- 11. Monge-Ampère equations
- 12. Symmetries and contact transformations of Monge-Ampère equations
- 13. Conservation laws
- 14. Monge-Ampère 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-Ampère 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.
