Book contents
- Frontmatter
- Contents
- Preface
- Part I Symmetries and Integrals
- Part II Symplectic Algebra
- Part III Monge–Ampère Equations
- Part IV Applications
- Part V Classification of Monge–Ampère equations
- 19 Classification of symplectic MAOs on two-dimensional manifolds
- 20 Classification of symplectic MAEs on two-dimensional manifolds
- 21 Contact classification of MAEs on two-dimensional manifolds
- 22 Symplectic classification of MAEs on three-dimensional manifolds
- References
- Index
19 - Classification of symplectic MAOs on two-dimensional manifolds
from Part V - Classification of Monge–Ampère equations
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- Part I Symmetries and Integrals
- Part II Symplectic Algebra
- Part III Monge–Ampère Equations
- Part IV Applications
- Part V Classification of Monge–Ampère equations
- 19 Classification of symplectic MAOs on two-dimensional manifolds
- 20 Classification of symplectic MAEs on two-dimensional manifolds
- 21 Contact classification of MAEs on two-dimensional manifolds
- 22 Symplectic classification of MAEs on three-dimensional manifolds
- References
- Index
Summary
The problem of equivalence and classification of Monge–Ampère equations goes back to Sophus Lie's papers from the 1870s and 1880s (see [65, 66, 67]). Sophus Lie have raised the following problem. Find equivalence classes of nonlinear second-order differential equations with respect to the group of contact transformations.
Sophus Lie himself had found conditions to transform a Monge–Ampère equation (MAE) to a quasi-linear one and to a linear equation with constant coefficients. The important steps in a solution of this problem were made by Darboux and Goursat [29], who had basically treated the hyperbolic Monge–Ampère equations.
As far as we know, a complete proof of Lie's theorems had never been published. The first results in this direction were obtained in [73, 74, 77].
In this part we consider the problem of local equivalence for Monge–Ampère equations and Monge–Ampère operators.
In [53, 54, 55, 59, 60, 61] it was shown that for Monge–Ampère equations of general type this problem can be reduced to the equivalence problem for e-structures. This leads to a solution of the equivalence problems and to a classification of Monge–Ampère equations and Monge–Ampère operators.
In this chapter we consider the problem of classification of symplectic Monge–Ampère operators of hyperbolic, elliptic and mixed types.
Let ω1 and ω2 be two effective differential 2-forms on the cotangent bundle T*M.
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- Contact Geometry and Nonlinear Differential Equations , pp. 385 - 421Publisher: Cambridge University PressPrint publication year: 2006