2 APPLICATIONS OF THE LOCAL GEOMETRY OF D IFFIETIES

The calculus on infinite jet spaces may be given a nice formalization with diffieties, which are infinite-dimensional smooth Frechet differe ntiable manifolds, equipped with Cartan distributions, i.e., finite-di mensional involutive distributions. Lie-Backlund morphisms between dif fieties are compatible with the Cartan distributions, We show that a l ocal Lie-Backlund fiber bundle may be endowed with a differential dime nsion, which parallels the differential transcendence degree of a diff erential field extension. The analogue of a differential transcendence basis is also introduced. Our first application leads to the number o f degrees of freedom of nonholonomic constraints without having recour se to the virtual displacements of nonholonomic mechanics, the meaning of which is still being debated. The second application shows that th e state-variable representation of a nonlinear control system may exhi bit derivatives of the input variables.