Contact geometry and nonlinear differential equations
著者
書誌事項
Contact geometry and nonlinear differential equations
（Encyclopedia of mathematics and its applications / edited by G.-C. Rota, v. 101）
Cambridge University Press, 2007
- タイトル別名
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Contact geometry and non-linear differential equations
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注記
Includes bibliographical references (p. 487-492) and index
内容説明・目次
内容説明
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).
目次
- 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.
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