Contact Geometry and Non-linear Differential Equations

✍ Scribed by Kushner, Alexei & Lychagin, Valentin & Rubtsov, Vladimir

• Publisher: Cambridge University Press
• Year: 2006
• Tongue: English
• Leaves: 518
• Series: Encyclopedia of Mathematics and Its Applications
• Category: Library

✦ Synopsis

The most relevant FirstSearch result that mentions Lie, Monge, and Ampère!
Kumei's 1981 dissertation mentions Lie, Legendre, and Ampère. (Monge-Ampère equations are invariant under Legendre transforms.)
mentions symplectic geometry:
* Frauenfelder, Urs, and Joa Weber. “The Fine Structure of Weber’s Hydrogen Atom: Bohr–Sommerfeld Approach.” Zeitschrift Für Angewandte Mathematik Und Physik 70, no. 4 (June 21, 2019): 105.

Cantwell PDF p. 148 gives a good definition of symplectic manifolds , in the context of classical mechanics, Hamilton's equations, and Poisson brackets. ( OED: " συμπλεκτικός twining or plaiting together, copulative")

The conserved elements of a Hamiltonian system … define a vector space. The rules of algebra in this space are given by the skew-symmetry of the composition operator (Poisson bracket) (4.43), the additive properties in (4.44), and the Jacobi identity (4.47). A vector space with these special properties is called a symplectic space, and the solution of the Hamiltonian system is said to lie on a symplectic manifold. This odd word comes from the greek symplektikos meaning “twining together,” from syn (together) and plekein (to twine). It is an apt description of the solution trajectories of a periodically forced Hamiltonian system, which can be visualized as a family of spiraling curves on a torus in a three-dimensional phase space where the third dimension is the phase angle of the forcing function.

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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).

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MR2352610
The aim of this book is to explain how methods from symplectic and contact geometry, and the use of the theory of symmetries, can be helpful in solving even highly nontrivial ordinary and partial differential equations without the need to have recourse to approximate numerical techniques or computer algebra. In doing so, the authors have accomplished a great job: the book excels in clarity and accessibility on the one hand, and in completeness on the other hand. The basic ideas and pioneering work of Lie and Cartan concerning the transformation of solving differential equations into a geometric or algebraic problem are treated and interpreted in a novel and modern way. In particular, it is shown that a large class of differential equations can be naturally embedded into the framework of symplectic or contact geometry so that one can apply, among other things, the whole machinery of the Hodge-de Rham calculus to it. Throughout, the authors seek geometric transparency of the ideas they expound upon and combine mathematical rigor with the search for exact solutions to nontrivial problems. Many examples, simple as well as advanced, serve to illustrate the theoretical part, and from the physical applications that are treated we may mention, for instance, an analysis of laser beams and the dynamics of cyclones.
The book is divided into five parts, each consisting of several chapters. Part I is devoted to the theory of symmetries and integrals of distributions and of ordinary differential equations on a manifold. First, a general discussion of distributions and their Lie algebra of symmetries is given. One of the highlights here is the Lie-Bianchi theorem, which gives a condition and a constructive algorithm for the integrability by quadratures of a distribution in terms of a solvable Lie algebra of symmetries. The treatment continues with, among others, the theories of symmetries of ordinary differential equations, linear symmetries of self-adjoint differential operators with application to the Schrödinger operator, reduction by symmetries and the Lie superposition principle.
Part II deals with symplectic algebra. It starts with a review of the basic elements from the theory of symplectic vector spaces and exterior algebra on symplectic vector spaces. Next, the symplectic classification of exterior $2$-forms in four and in higher dimensions and the symplectic classification of $3$-forms in six dimensions are treated.
Part III is then devoted to the study of the Monge-Ampère equations. After a general discussion of symplectic and contact manifolds, Monge-Ampère differential operators are introduced. Symmetries and contact transformations, as well as conservation laws of the Monge-Ampère equations, are treated. Some geometric structures related to Monge-Ampère equations on a two-dimensional manifold are described. This part ends with a treatment of first-order partial differential equations in two dimensions, namely the class of Jacobi equations.
Part IV is completely devoted to physical applications of the theory, which are taken from nonlinear acoustics (symmetries, conservation laws and exact solutions of the Khokhlov-Zabolotskaya equation, which describes the propagation of a sound beam in a nonlinear medium), nonlinear thermal conductivity (symmetries and invariant solutions of a version of the Kolmogorov-Petrovskiĭ-Piskunov equation with nonlinear diffusion) and also from meteorology (Monge-Ampère equations in semi-geostrophic models), which is based upon some ongoing research by one of the authors with I. Roulstone from the University of Surrey (UK).
Finally, Part V is devoted to the classification problem of Monge-Ampère equations in the framework of symplectic and contact geometry for which the most complete solution exists in two and three dimensions. The two-dimensional case is related to the classical equivalence problem of Sophus Lie, and a modern version of Lie's results are presented. A complete proof of these results was in fact first obtained by the authors of this book in some papers from the 1980's. Reviewed by Frans Cantrijn

✦ Subjects

Contact manifolds, Differential equations, Nonlinear, Differential equations Nonlinear, Variétés de contact (Géométrie) Équations différentielles non linéaires, MATHEMATICS -- Geometry -- General, Meetkunde, Niet-lineaire analyse