Contact Geometry and Nonlinear Differential Equations

Shows novel and modern ways of solving differential equations using methods from contact and symplectic geometry.

Alexei Kushner (Author), Valentin Lychagin (Author), Vladimir Rubtsov (Author)

9780521824767, Cambridge University Press

Hardback, published 21 December 2006

518 pages, 58 b/w illus. 30 tables
24.1 x 16.2 x 3.2 cm, 0.864 kg

'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

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).

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.

Subject Areas: Topology [PBP], Geometry [PBM], Differential calculus & equations [PBKJ]