title: Applications Of Contact Geometry And Topology In Physics
author: Arkady L Kholodenko
description: Although contact geometry and topology is briefly discussed in V I Arnol'd's book "Mathematical Methods of Classical Mechanics "(Springer-Verlag, 1989, 2nd edition), it still remains a domain of research in pure mathematics, e.g. see the recent monograph by H Geiges "An Introduction to Contact Topology" (Cambridge U Press, 2008). Some attempts to use contact geometry in physics were made in the monograph "Contact Geometry and Nonlinear Differential Equations" (Cambridge U Press, 2007). Unfortunately, even the excellent style of this monograph is not sufficient to attract the attention of the physics community to this type of problem. This book is the first serious attempt to change the existing status quo. In it we demonstrate that, in fact, all branches of theoretical physics can be rewritten in the language of contact geometry and topology: from mechanics, thermodynamics and electrodynamics to optics, gauge fields and gravity; from physics of liquid crystals to quantum mechanics and quantum computers, etc. The book is written in the style of famous Landau Lifshitz (L L) multivolume course in theoretical physics. This means that its readers are expected to have solid background in theoretical physics (at least at the level of the L L course). No prior knowledge of specialized mathematics is required. All needed new mathematics is given in the context of discussed physical problems. As in the L L course some problems/exercises are formulated along the way and, again as in the L L course, these are always supplemented by either solutions or by hints (with exact references). Unlike the L L course, though, some definitions, theorems, and remarks are also presented. This is done with the purpose of stimulating the interest of our readers in deeper study of subject matters discussed in the text.
zlib_text: Contact Geometry and Nonlinear Differential Equations
  Methods from contact and symplectic geometry can be used to solve highly non-trivial
  non-linear 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–de Rham calculus can be applied. In this way a wide class of
  equations can be tackled, including quasi-linear equations, Monge–Ampère equations
  (which play an important role in modern theoretical physics and meteorology).
  The main features of the boo