A geometric point of view on implicit systems of differential algebraic equations

We present some work done in relation with the ANR LEDA project. Classically an implicit system of differential algebraic equations defines a variety in a jet-space. Supposing that this variety is smooth enough, one can define a direction field at any of its regular points using its tangent plane and the so-called contact plane at this point. Differential algebraic equations (DAEs) are a general form of differential equations that are not completely solvable for the derivatives of all components of their states variables. The talk presents the relationships between the above classical framework and a class of DAEs occuring in practice. We show how an explicit ODE equivalent to a given DAE can be found using some Lie point symmetry of the associated variety in a jet-space.