Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction and summary
- 2 PDE systems, pfaffian systems and vector field systems
- 3 Cartan's local existence theorem
- 4 Involutivity and the prolongation theorem
- 5 Drach's classification, second order PDEs in one dependent variable, and Monge characteristics
- 6 Integration of vector field systems V satisfying dim V' = dim V + 1
- 7 Higher order contact transformations
- 8 Local Lie groups
- 9 Structural classification of 3-dimensional Lie algebras over the complex numbers
- 10 Lie equations and Lie vector field systems
- 11 Second order PDEs in one dependent and two independent variables
- 12 Hyperbolic PDEs with Monge systems admitting two or three first integrals
- 13 Classification of hyperbolic Goursat equations
- 14 Cartan's theory of Lie pseudogroups
- 15 The equivalence problem
- 16 Parabolic PDEs and associated PDE systems
- 17 The equivalence problem for general 3-dimensional pfaffian systems in five variables
- 18 Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
- Bibliography
- Index
15 - The equivalence problem
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Preface
- 1 Introduction and summary
- 2 PDE systems, pfaffian systems and vector field systems
- 3 Cartan's local existence theorem
- 4 Involutivity and the prolongation theorem
- 5 Drach's classification, second order PDEs in one dependent variable, and Monge characteristics
- 6 Integration of vector field systems V satisfying dim V' = dim V + 1
- 7 Higher order contact transformations
- 8 Local Lie groups
- 9 Structural classification of 3-dimensional Lie algebras over the complex numbers
- 10 Lie equations and Lie vector field systems
- 11 Second order PDEs in one dependent and two independent variables
- 12 Hyperbolic PDEs with Monge systems admitting two or three first integrals
- 13 Classification of hyperbolic Goursat equations
- 14 Cartan's theory of Lie pseudogroups
- 15 The equivalence problem
- 16 Parabolic PDEs and associated PDE systems
- 17 The equivalence problem for general 3-dimensional pfaffian systems in five variables
- 18 Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge
- Bibliography
- Index
Summary
In chapters 5 and 11 we have seen that any second order PDE S ⊂ J2 admits two Monge characteristic vector field systems. S is said to be hyperbolic if these are different, and parabolic if they coincide. Vessiot's study of those hyperbolic PDEs for which each Monge system admits at least two functionally independent first integrals led to a remarkable classification, presented in chapters 12 and 13.
The corresponding classification of those parabolic PDEs for which the double Monge system admits at least two functionally independent first integrals is the subject of Cartan's ‘five variable paper’ [Cartan 1910].
While Vessiot's classification is based upon a straightforward stepwise simplification of the Monge systems, Cartan instead starts by solving a very general equivalence problem, which he then applies to the special case of parabolic PDEs.
Cartan's ideas for solving the equivalence problem were first announced in the Comptes Rendus note [Cartan 1902]. In order to make this comprehensible he then went on with his general theory of Lie pseudogroups in [Cartan 1904] and [Cartan 1905], whereupon the first chapter of [Cartan 1908] is devoted to a full explanation of the equivalence problem. And this is then applied to the classification of parabolic PDEs in [Cartan 1910].
This chapter is a survey of Cartan's solution of the equivalence problem, mainly following [Cartan 1937a]. For more modern references see e.g. [Gardner 1989], [Morimoto 1993], and [Olver 1995].
A different and very interesting approach to the equivalence problem has been developed by Pommaret, taking [Vessiot 1903] as point of departure—see e.g. [Pommaret 1978] and [Pommaret 1983].
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- Information
- Lie's Structural Approach to PDE Systems , pp. 421 - 441Publisher: Cambridge University PressPrint publication year: 2000