# Solving ODE via contact geometry

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential equations. I am aware that he refers to V.I. Arnold's "Geometrical Methods in the Theory of ODEs" for details, but couldn't find a satisfactory example.

I am looking for a concrete example where a contact geometric point of view helps in answering a question about an ODE.

## 2 Answers

A comprehensive reference, with many worked-out examples, is Contact Geometry and Nonlinear Differential Equations (2007). Here is a review.

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).

The concrete application that might be what you are looking for is to the Knizhnik–Zamolodchikov equation (which models the propagation of sound beams in a non-linear medium).

• This goes precisely in the direction I was hoping for. Are you aware of a self contained, short example? – Lukas Salchow Sep 3 '17 at 16:41

The thesis of Raouf Dridi contains several worked examples of the application of Lie and Cartan's methods to ODEs.

• I don't see that Dridi's approach uses contact geometry. – Ben McKay Sep 5 '17 at 9:51