Contact Coordinates for Partial Prolongations

Let V be a vector field distribution on manifold M . We give an efficient algorithm for the construction of local coordinates on M such that V may be locally expressed as some partial prolongation of the contact distribution C (1) q , on the 1 st order jet bundle of maps from R to R, q ≥ 1. It is proven that if V is locally equivalent to a partial prolongation of C (1) q then the explicit construction of contact coordinates algorithmically depends upon the determination of certain first integrals in a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M . The number of these first integrals that must be computed satisfies a natural minimality criterion. These results therefore provide a full and constructive generalisation of the classical Goursat normal form from the theory of exterior differential systems. 2000 MSC: 58J60, 34H05, 93B18, 93B27. keywords: Generalised Goursat normal form, partial prolongation, algorithm, nonlinear control theory, contact coordinates, Brunovsky normal form