The classic training of an engineer or scientist often begins with the study of
kinematics, where the concept of a Velocity vector in introduced as the limit of
the derivative of a position vector R with respect to time. It is little
appreciated that this primitive idea is a topological constraint on the domain
of variables, {R,V,t} that is best written as dRVdt = 0. In this later format
it is apparent that he space of variables is constrained by a Cartan exterior
differential system consisting of N vanishing 1forms. On the otherhand if an initial space, R0, at parametric value t=t0, is mapped to a final space, R, at parametric value t=t, then given the map it is possible to deduce the relations
In the more general situation, the fundamental equation is dRVdt = (an Error or
flucutation 1form) <> 0. The Error 1forms can be interpreted as
fluctuations which may have zero values on average, or are closed in a
topological limit sense (which implies that each 1form of constraint has a
vanishing exterior derivative and is integrable). Other constraints are
possible. The fluctuation 1forms need not be integrable in the Frobenius sense.
Hence the evolutionary process can not be represented by a single parameter
group. Phenomenologically, dissipation is often attributed to such anholonomic
fluctuations.
