 
Summary: First steps of local contact algebra
V.I.Arnold
Dedicated to H.S.M. Coxeter
1 Introduction
The belief that all simple (having no continuous moduli) objects in the nature are controlled
by the Coxeter groups is a kind of religion. The corresponding theorem in singularity theory
is due to A.B.Givental [1]. It classies simple singularities of caustics and wave fronts,
dened by the projections of Lagrange and Legendre subvarieties of symplectic and contact
manifolds, in terms of the Coxeter euclidean re
ections groups, extending to the case of
singular varieties my previous A D E { classication [2] (corresponding to smooth
submanifolds).
The present work is an attempt to start the classication of singular simple curves in
contact manifolds.
The idea that every reasonable mathematical theory should have a symplectic and con
tact versions is also based on the success of Coxeter's extension of linear algebra (considered
as the theory of the root systems A) to other mirrors congurations. The application of
this idea to calculus has led to the foundation of symplectic and contact topologies (see
[3]).
In the present article the same idea is applied to a modest local problem. It is astonishing
that this problem { the classication of simple curves in a contact space { is rather diĘcult
