Noncommutative flows I: dynamical invariants

We show that a noncommutative dynamical system of the type that occurs in quantum theory can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. The nature of these dynamical principles is similar to that of the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a ``Riemannian metric", and a potential. These structures are classified in terms of geometric objects which, in the simplest cases, occur in finite dimensional matrix algebras. As a consequence, we obtain a new classification of E_0-semigroups acting on type I factors.