  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.
