On symplectization of 1-jet space and differential invariants of point pseudogroup

Abstract

In this paper we suggest an approach to the study of the action of point pseudogroup in (contact) 1-jet space J 1 R n . This approach is based on the following idea. We replace the canonic projection π 1 , 0 : J 1 R n → J 0 ( R n ) from 1-jet space to 0-jet space with projective action on the fiber by some bundle, which is called symplectization , with the linear action on the fiber. This makes it possible to apply methods of studying the linear actions of algebraic groups and to find the set of independent differential invariants. Finally, we rewrite the action of point pseudogroup in terms of these invariants and classify the orbits of point pseudogroup action. We also give two examples of application of this method. In the first example we obtain classification of contact vector fields in J 1 R n , and in the second we obtain classification of symbols of linear differential operators.