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Contact Geometry and Nonlinear Differential Equations

Contact Geometry and Nonlinear Differential Equations

$204.00 (C)

Part of Encyclopedia of Mathematics and its Applications

  • Date Published: January 2007
  • availability: Available
  • format: Hardback
  • isbn: 9780521824767

$ 204.00 (C)
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  • 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).

    • Accessible and useful for both experts and non-specialists
    • Many new ideas, first time available in book form
    • Methods applicable to real world applications
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    Reviews & endorsements

    "The book excels in clarity and accessibility on the one hand, and in completeness on the other hand."
    Frans Cantrijn, Mathematical Reviews

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    Product details

    • Date Published: January 2007
    • format: Hardback
    • isbn: 9780521824767
    • length: 518 pages
    • dimensions: 241 x 162 x 32 mm
    • weight: 0.864kg
    • contains: 58 b/w illus. 30 tables
    • availability: Available
  • 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.

  • Authors

    Alexei Kushner, Astrakhan State Pedagogical University
    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, Universitetet i Tromsø, Norway
    Valentin Lychagin is a Professor at the Institute of Mathematics and Statistics, Tromsø University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow.

    Vladimir Rubtsov, Université d'Angers, France
    Vladimir Rubtsov is a Professor at the Département de Mathématiques, Angers University, and a Senior Researcher at the Institute for Theoretical and Experimental Physics in Moscow.

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