Abstract: Noncommutative flows I

\abstract
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 \esg s acting on type $I$ factors.
\endabstract