A constructive generalised Goursat normal form

We provide necessary and sufficient conditions on the derived type of a vector field distribution VV in order that it be locally equivalent to a partial prolongation of the contact distribution Cq(1), on the 1st order jet bundle of maps from RR to RqRq, q⩾1q⩾1. This result fully generalises the Goursat normal form from the theory of exterior differential systems. Our proof is constructive: it is proven that if VV is locally equivalent to a partial prolongation of Cq(1) then the explicit construction of contact coordinates algorithmically depends upon the integration of a sequence of geometrically defined and algorithmically determined integrable Pfaffian systems on M. Though the tools required are rather different, our main theorem may be regarded as a generalisation of the work of R.B. Gardner and W.F. Shadwick on the feedback linearisation of autonomous nonlinear control systems.